LIBRARY
UNIVERSITY OF CALIFORNIA.
Deceived /fa/ OSS , i8g</
f *
tAcce&sion No. *7 ft
£0 .
OF
TTNIVERSI'
ABEL'S THEOEEM AND THE ALLIED THEOKY
INCLUDING THE
THEOKY OF THE THETA FUNCTIONS
: C. J. CLAY AND SONS,
CAMBRIDGE UNIVERSITY PRESS WAREHOUSE,
AVE MARIA LANE.
OSIaggoto : 263, ARGYLE STREET.
P. A. BROCKHAUS.
Hork: THE MACMILLAN COMPANY.
ABEL'S THEOREM
AND THE
ALLIED THEOEY
INCLUDING THE THEORY OF THE
THETA FUNCTIONS
OF THE
UNIVERSITY
BY
H. F. BAKER, M.A.
FELLOW AND LECTURER OF ST JOHN'S COLLEGE,
UNIVERSITY LECTURER IN MATHEMATICS.
CAMBRIDGE:
AT THE UNIVERSITY PRESS.
1897
[All Rights reserved]
PRINTED BY J. AND C. F. CLAY,
AT THE UNIVERSITY PRESS.
To 4
^w-
PREFACE.
IT may perhaps be fairly stated that no better guide can be found to the
analytical developments of Pure Mathematics during the last seventy years
than a study of the problems presented by the subject whereof this volume
treats. This book is published in the hope that it may be found worthy to
form the basis for such study. It is also hoped that the book may be
serviceable to those who use it for a first introduction to the subject.
And an endeavour has been made to point out what ^are conceived to be the
most artistic ways of formally developing the theory regarded as complete.
The matter is arranged primarily with a view to obtaining perfectly
general, and not merely illustrative, theorems, in an order in which they can
be immediately utilised for the subsequent theory; particular results, however
interesting, or important in special applications, which are not an integral
portion of the continuous argument of the book, are introduced only so far
as they appeared necessary to explain the general results, mainly in the
examples, or are postponed, or are excluded altogether. The sequence and
scope of ideas to which this has led will be clear from an examination of the
table of Contents. fc-
The methods of Riemann, as far as they are explained in books on the
general theory of functions, are provisionally regarded as fundamental ; but
precise references .are given for all results assumed, and great pains have
been taken, in the theory of algebraic functions and their integrals, and in
the analytic theory of theta functions, to provide for alternative developments
of the theory. If it is desired to dispense with Riemann's existence theorems,
the theory of algebraic functions may be founded either on the arithmetical
ideas introduced by Kronecker and by Dedekind and Weber ; or on the
quasi-geometrical ideas associated with the theory of adjoint polynomials ;
while in any case it does not appear to be convenient to avoid reference to
either class of ideas. It is believed that, save for some points in the
periodicity of Abelian integrals, all that is necessary to the former ele
mentary development will be found in Chapters IV. and VII., in connection
with which the reader may consult the recent -paper of Hensel, Acta
Mathematica, xvm. (1894), and also the papers of Kronecker and of
B. • b
V] PREFACE.
Dedekind and Weber, Grelle's Journal, xci., xcn. (1882). And it is hoped
that what is necessary for the development of the theory from the elemen
tary geometrical point of view will be understood from Chapter VI., in
connection with which the reader may consult the Abel'sche Functionen of
Clebsch and Gordan (Leipzig, 1866) and the paper of Noether, Mathematische
Annalen, vii. (1873). In the theory of Riemann's theta functions, the
formulae which are given relatively to the £ and g>- functions, and the
general formulae given near the end of Chapter XIV., will provide sufficient
indications of how the theta functions can be algebraically denned ; the
reader may consult Noether, Mathematische Annalen, xxxvn. (1890), and
Klein and Burkhardt, ibid. xxxn. — xxxvi. In Chapters XV., XVII., and
XIX., and in Chapters XVIII. and XX., are given the beginnings of that
analytical theory of theta. functions from which, -in conjunction with the
general theory of functions of several independent variables, so much is to
be hoped ; the latter theory is however excluded from this volume.
To the reader who does not desire to follow the development of this
volume consecutively through, the following course may perhaps be sug
gested; Chapters I., II., III. (in part), IV., VI. (to § 98), VIII., IX., X.,
XL (in part), XVIII. (in part), XII. , XV. (in. part); it is also possible to
begin with the analytical theory of theta functions, reading in order Chapters
XV., XVI., XVII., XIX., XX.
The footnotes throughout the volume are intended to contain the
mention of all authorities used in its preparation ; occasionally the hazardous
plan of adding to the lists of references during the passage of the sheets
through the press, has been adopted ; for references omitted, and for refer
ences improperly placed, only mistake can be pleaded. Complete lists of
papers are given in the valuable report of Brill and Noether, " Die Entwicklung
der Theorie der algebraischen Functionen in alterer und neuerer Zeit,"
Jahresbericht der Deutschen Mathematiker-Vereinigung, Dritter Band, 1892 — 3
(Berlin. Reimer, 1894); this report unfortunately appeared only after the
first seventeen chapters of this volume, with the exception of Chapter XL,
and parts of VIL, were in manuscript ; its plan is somewhat different from
that of this volume, and it will be of advantage to the reader to consult
it. Other books which have appeared during the progress of this volume, too
late to effect large modifications, have not been consulted. The examples
throughout the volume are intended to serve several different purposes ; to
provide practice in the ideas involved in the general theory ; to suggest the
steps of alternative developments without interrupting the line of reasoning
in the text; and to place important consequences which are not utilised, if
at all, till much subsequently, in their proper connection.
For my first interest in the subject of this volume, I desire to acknowledge
my obligations to the generous help given to me during Gottingen vacations,
PREFACE. Vll
on two occasions, by Professor Felix Klein. In the preparation of the book
I have been largely indebted to his printed publications ; the reader is
recommended to consult also his lithographed lectures, especially the one
dealing with Riemann surfaces. In the final revision of the sheets in
their passage through the press, I have received help from several friends.
Mr A. E. H. Love, Fellow and Lecturer of St John's College, has read
the proofs of the volume ; in the removal of obscurities of expression
and in the correction of press, his untiring assistance has been of great
value to me. Mr J. Harkness, Professor of Mathematics at Bryn Mawr
College, Pennsylvania, has read the proofs from Chapter XV. onwards; many
faults, undetected by Mr Love or myself, have yielded to his perusal ; and
I have been greatly helped by his sympathy in the subject-matter of the
volume. To both these friends I am under obligations not easy to discharge.
My gratitude is also due to Professor Forsyth for the generous interest he
has taken in the book from its commencement. While, it should be added,
the task carried through by the Staff of the University Press deserves more
than the usual word of acknowledgment.
This book has a somewhat ambitious aim ; and it has been written under
the constant pressure of other work. It cannot but be that .many defects
will be found in it. But the author hopes it will be sufficient to shew that
the subject offers for exploration a country of which the vastness is equalled
by the fascination.
ST JOHN'S COLLEGE, CAMBRIDGE.
April 26, 1897.
CONTENTS.
CHAPTER I.
THE SUBJECT OF INVESTIGATION.
§§ PAGES
1 Fundamental algebraic irrationality 1
2, 3 The places and infinitesimal on a Riemann surface . . . 1, 2
4, 5 The theory unaltered by rational transformation . .' 3 — 6
6 The invariance of the deficiency in rational transformation ; if a
rational function exists of order 1, the surface is of zero
deficiency ........... 7, 8
7, 8 The greatest number of irremoveable parameters is 3p - 3 . . 9, 10
9, 10 The geometrical statement of the theory 11, 12
11 Generality of Riemann's methods 12, 13
CHAPTER II.
THE FUNDAMENTAL FUNCTIONS ON A RlEMANN SURFACE.
12 Riemann's existence theorem provisionally regarded as fundamental 14
13 Notation for normal elementary integral of second kind . . 15
14 Notation for normal elementary integral of third kind ... 15
15 Choice of normal integrals of the first kind 16
16 Meaning of the word period. General remarks . . . ^. 16, 17
17 Examples of the integrals, and of the places of the surface . 18 — 20
18 Periods of the normal elementary integrals of the second kind . 21
19 The integral of the second kind arises by differentiation from the
integral of the third kind 22, 23
20 Expression of a rational function by integrals of the second kind . 24
21 Special rational functions, which are invariant in rational trans
formation . 25, 26
22 Riemann normal integrals depend on mode of dissection of the
surface 26
CHAPTER III.
THE INFINITIES OF RATIONAL FUNCTIONS.
23 The interdependence of the poles of a rational function . . 27
24, 25 Condition that specified places be the poles of a rational function . 28 — 30
26 General form of Weierstrass's gap theorem 31, 32
27 Provisional statement of the Riemann-Roch theorem ... 33, 34
K CONTENTS.
§§ PAGES
28, 29 Cases when the poles coalesce ; the p critical integers . . 34, 35
30 Simple anticipatory geometrical illustration ...... 36, 37
31 — 33 The (p-l)p(p + l) places which are the poles of rational functions •
of order less than p + l 38 — 40
34 — 36 There are at least 2jo + 2 such places which are distinct . . 41 — 44
37 Statement of the Riemann-Roch theorem, with examples . . 44 — 46
CHAPTER IV.
SPECIFICATION OF A GENERAL FORM OF RIEMANN'S INTEGRALS.
38 Explanations in regard to Integral Rational Functions . . 47, 48
39 Definition of dimension ; fundamental set of functions for the
expression of rational functions 48 — 52
40 Illustrative example for a surface of four sheets . . . . 53, 54
41 The sum of the dimensions of the fundamental set of functions
is p + n-l 54, 55
42 Fundamental set for the expression of integral functions . . 55, 56
43 Principal properties of the fundamental set of integral functions . 57 — 60
44 Definition of derived set of special functions 00, 0j, ..., 4>n_l . 61 — 64
45 Algebraical form of elementary integral of the third kind, whose
infinities are ordinary places ; and of integrals of the first
kind . . . . 65 — 68
46 Algebraical form of elementary, integral of the third kind in general 68 — 70
47 Algebraical form of integral of the second kind, independently
deduced 71—73
48 The discriminant of the fundamental set of integral functions . 74
49 Deduction of the expression of a certain fundamental rational
function in the general case 75 — 77
50 The algebraical results of this chapter are sufficient to replace
Riemann's existence theorem * . 78, 79
CHAPTER V.
CERTAIN FORMS OF THE FUNDAMENTAL EQUATION OF THE RIEMANN SURFACE.
51 Contents of the chapter 80
52 When p>l, existence of rational function of the second order
involves a (1, 1) correspondence 81
53—55 Existence of rational function of the second order involves the
hyperelliptic equation 81 — 84
56, 57 Fundamental integral functions and integrals of the first kind . 85 — 86
58 Examples 87
59 Number of irremoveable parameters in the hyperelliptic equation ;
transformation to the canonical form 88 — 89
60—63 Weierstrass's canonical equation for any deficiency . . . 90—92
CONTENTS.
XI
§§
64—66
67, 68
69—71
72—79
Actual formation of the equation . . .
Illustrations of the theory of integral functions for Weierstrass's
canonical equation
The method can be considerably generalised . . • . . • .
Hensel's determination of the fundamental integral functions
PAGES
93—98
99—101
102—104
105—112
CHAPTER VI.
GEOMETRICAL INVESTIGATIONS.
80 Comparison of the theory of rational functions with the theory
of intersections of curves . . . . ... . 113
81 — 83 Introductory indications of elementary form of theory . . . 113—116
84 The method to be followed in this chapter 117
85 Treatment of infinity. Homogeneous variables might be used . 118,- 119
86 Grade of an integral polynomial ; number of terms ; generalised
zeros . 120, 121
87 Adjoint polynomials ; definition of the index of a singular place . 122
88 Pliicker's equations ; connection with theory of discriminant • . 123, 124
89, 90 Expression of rational functions by adjoint polynomials . . . 125, 126
91 Expression of integral of the first kind . . . . . 127
92 Number of terms in an adjoint polynomial ; determination of
elementary integral of the third kind . . ... . 128 — 132
93 Linear systems of adjoint polynomials ; reciprocal theorem . . 133, 134
94, 95 Definitions of set, lot, sequent, equivalent sets, coresidual sets . 135
96, 97 Theorem of coresidual sets ; algebraic basis of the theorem . . 136
98 A rational function of order less than p + 1 is expressible by <£-
polynomials , 137
99, 100 Criticism of the theory; Cayley's theorem . .' ' .'"', , . 138—141
101 — 104 Rational transformation by means of (^-polynomials . . . 142—146
105 — 108 Application of special sets 147 — 151
109 The hyperelliptic surface ; transformation to canonical form . 152
1.10 — 114 Whole rational theory can be represented by means of the invari
ant ratios of (^-polynomials ; number of relations connecting
these 153—159
115 — 119 Elementary considerations in regard to curves in space . . 160 — 167
CHAPTER VII.
COORDINATION OF SIMPLE ELEMENTS. TRANSCENDENTAL UNIFORM
FUNCTIONS.
Scope of the chapter 168
Notation for integrals of the first kind . . . . . 169
The function ^ (x, a; z, cl5 ..., cp) expressed by Riemann integrals 170, 171
Derivation of a certain prime function 172
Applications of this function to rational functions and integrals 173
Xll CONTENTS.
§§ PAGES
126—128 The function ^(x,a-, z, c) ; its utility for the expression of
rational functions 174 — 176
129 The derived prime function E(x,z); used to express rational
functions 177
130, 131 Algebraic expression of the functions ^ (x, a ; z, clt ...,cp),
ty{x, a; z,c) 177, 178
132 Examples of these functions; they determine algebraic expres
sions for the elementary integrals 179 — 182
133, 134 Derivation of a canonical integral of the third kind; for which
interchange of argument and parameter holds; its algebraic
expression ; its relation with Riemann's elementary normal
integral 182—185
135 Algebraic theorem equivalent to interchange of argument and
parameter 185
136 Elementary canonical integral of the second kind . . . 186, 187
137 Applications. Canonical integral of the third kind deduced from
the function ^(.v,a; z,c^ ...,cp). Modification for the func
tion ty(x, a; z, c) 188—192
138 Associated integrals of first and second kind. Further canonical
integrals. The algebraic theory of the hyperelliptic integrals
in one formula. . . 193, 194
139, 140 Deduction of Weierstrass's and Riemann's relations for periods
of integrals of the first and second kind .... 195 — 197
141 Either form is equivalent to the other 198
142 Alternative proofs of Weierstrass's and Riemann's period relations 199, 200
143 ' Expression of uniform transcendental function by the function
ty(x, a; z, c) ' . . . . 201
144, 145 Mittag-Lefner's theorem . . . ... . . 202 204
146 Expression of uniform transcendental function in prime factors 205
147 General form of interchange of argument and parameter, after
Abel 206
CHAPTER VIII.
ABEL'S THEOREM. ABEL'S DIFFERENTIAL EQUATIONS.
148—150 Approximative description of Abel's theorem 207—210
151 Enunciation of the theorem 210
152 The general theorem reduced to two simpler theorems . . 211, 212
153, 154 Proof and analytical statement of the theorem .... 212 214
155 Remark; statement in terms of polynomials . . . . 215
156 The disappearance of the logarithm on the right side of the
equation . . 216
157 Applications of the theorem. Abel's own proof .... 217 222
158, 159 The number of algebraically independent equations given by the
theorem. Inverse of Abel's theorem 222 224
160, 161 Integration of Abel's differential equations " 225 231
162 Abel's theorem proved quite similarly for curves in space . . 231 — 234
CONTENTS. Xlll
CHAPTER IX.
JACOBI'S INVERSION PROBLEM.
§§ PAGES
163 Statement of the problem 235
164 Uniqueness of any solution 236
165 The necessity of using congruences and not equations . . 237
166, 167 Avoidance of functions with infinitesimal periods . . . 238, 239
168, 169 Proof of the existence of a solution 239—241
170 — 172 Formation of functions with which to express the solution;
connection with theta functions . 242—245
CHAPTER X.
RIEMANN'S THETA FUNCTIONS. GENERAL THEORY.
173 Sketch of the history of the introduction of theta functions . 246
174 Convergence. Notation. Introduction of matrices . . . 247, 248
175, 176 Periodicity of the theta functions. Odd and even functions . 249 — 251
177 Number of zeros is p '. 252
178 Position of the zeros in the simple case . . . i 'orf3 253, 254
179 The places TOI} ..., mp 255
180 Position of the zeros in general 256, 257
181 Identical vanishing of the theta functions ..... 258, 259
182, 183 Fundamental properties. Geometrical interpretation of the places
m1,...,mp • . . . . 259—267
184 — 186 Geometrical developments; special inversion problem; contact
curves • ., tj 268 — 273
187 Solution of Jacobi's inversion problem by quotients of theta
functions 274, 275
188 Theory of the identical vanishing of the theta function. Ex
pression of (^-polynomials by theta functions . . . 276 — 282
189—191 General form of theta function. Fundamental formulae. Periodicity 283 — 286
192 Introduction of the f functions. Generalisation of an elliptic formula 287
193 Difference of two f functions expressed by algebraic integrals and
rational functions ....... 288
194 — 196 Development. Expression of single f function by algebraic integrals 289 — 292
197, 198 Introduction of the $ functions. Expression by rational functions 292-295
CHAPTER XI.
THE HYPERELLIPTIC CASE OF RlEMANN'S THETA FUNCTIONS.
199 Hyperelliptic case illustrates the general theory .... 296
200 The places »i1>t.., mp. The rule for half periods . . . 297, 298
201, 202 Fundamental set of characteristics defined by branch places . 299—301
XIV
CONTENTS.
§§ PAGES
203 Notation. General theorems to be illustrated .... 302
204, 205 Tables in illustration of the general theory 303—309
206 — 213 Algebraic expression of quotients of hyperelliptic theta functions.
Solution of hyperelliptic inversion problem . . . . 309 — 317
214, 215 Single £ function expressed by algebraical integrals and rational
functions 318 — 323
216 Rational expression of $> function. Relation to quotients of theta
functions. Solution of inversion problem by g> function . . 323 — 327
217 Rational expression of $> function 327 — 330
218 — 220 Algebraic deduction of addition equation for theta functions
when p = 2; generalisation of the equation tr (u+v) a- (u-v)
= cr2w. o-V(^v-jptt) 330—337
221 Examples for the case p = 2. Qopel's biquadratic relation . . 337 — 342
CHAPTER XII.
A PARTICULAR FORM OF FUNDAMENTAL SURFACE.
222 Chapter introduced as a change of independent variable, and as
introducing a particular prime function .... 343
223—225 Definition of a group of substitutions ; fundamental properties . 343—348
226, 227 Convergence of a series ; functions associated with the group . 349 — 352
228 — 232 The fundamental functions. Comparison with foregoing theory
of this volume 353 — 359
233 — 235 Definition and periodicity of the Schottky prime function . . 359 — 364
236, 237 Its connection with the theta functions 364 — 366
238 A further function connected therewith 367 — 372
239 The hyperelliptic case . . . . .. . «. . . 372, 373
CHAPTER XIII.
RADICAL FUNCTIONS.
240 Introductory . . 374
241, 242 Expression of any radical function by Riemann's integrals, and
by theta functions 375, 376
243 Radical functions are a generalisation of rational functions . 377
244, 245 Characteristics of radical functions . . . . . . 378 — 381
246 — 249 Bitangents of a plane quartic curve 381 — 390
250, 251 Solution of the inversion problem by radical functions . . 390 — 392
CHAPTER XIV.
FACTORIAL FUNCTIONS.
252 Statement of results obtained. Notations 393, 394
253 Necessary dissection of the Riemann surface .... 395
254 Definition of a factorial function (including radical function).
Primary and associated systems of factorial functions . . 396, 397
CONTENTS.
XV
§§ PAGES
255 Factorial integrals of the primary and associated systems . . 397, 398
256 Factorial integrals which are everywhere finite, save at the fixed
infinities. Introduction of the numbers or, <r + 1 . . . 399
257 When <r + l>0, there are o- + l everywhere finite factorial functions
of the associated system ........ 400
258 Alternative investigation of everywhere finite factorial functions
of the associated system. Theory divisible according to the
values of o- + l and o-' + l 401, 402
259 Expression of these functions by everywhere finite integrals . 403
260 General consideration of the periods of the factorial integrals . 404
261, 262 Riemanri-Roch theorem for factorial functions. When or' + 1=0,
least number of arbitrary poles for fimction of the primary
system is or' + l 405, 406
263 Construction of factorial function of the primary system with
or' + l arbitrary poles . . . . . . . . 406, 407
264, 265 Construction of a factorial integral having only poles. Least
number of such poles, for an integral of the primary system,
is o- + 2 407, 410
266 This factorial integral can be simplified, in analogy with Riemann's
elementary integral of the second kind 411
267 Expression of the factorial function with or' + l poles in terms of
the factorial integral with o- + 2 poles. The factorial function
in analogy with the function i\r (x, a; z, clt ..., cp). . . 411 — 413
268 The theory tested by examination of a very particular case . 413 — 419
269 The radical functions as a particular case of factorial functions 419, 420
270 Factorial functions whose factors are any constants, having no
essential singularities ........ 421
271, 272 Investigation of a general formula connecting factorial functions
and theta functions 422 — 426
273 Introduction of the Schottky-Klein prime form, in a certain shape 427 — 430
274 Expression of a theta function in terms of radical functions, as '
a particular case of § 272 . . . . '-. . " . 430
275, 276 The formula of § 272 for the case of rational functions . . 431—433
277 The formula of § 272 applied to define algebraically the hyper-
elliptic theta function, and its theta characteristic . . 433 — 437
278 Expression of any factorial function by simple theta functions ;
examples 437, 433
279 Connection of theory of factorial functions with theory of auto-
morphic forms . 439 442
CHAPTER XV.
RELATIONS CONNECTING PRODUCTS OF THETA FUNCTIONS — INTRODUCTORY.
280
281
443
Plan of this and the two following chapters ....
A single-valued integral analytical function of p variables, which
is periodic in each variable alone, can be represented by a
series of exponentials .... 443 445
XVI CONTENTS.
§§ PAGES
282, 283 Proof that the 22p theta functions with half-integer character
istics are linearly independent ...... 446 — 447
284, 285 Definition of general theta function of order r ; its linear expres
sion by r1' theta functions. Any p-f2 theta functions of
same order, periods, and characteristic connected by a homo
geneous polynomial relation ....... 447 — 455
286 Addition theorem for hyperelliptic theta functions, or for the
general case when p<4 ........ 456 — 461
286, 288 Number of linearly independent theta functions of order r which
are all of the same parity ....... 461 — 464
289 Examples. The Gopel biquadratic relation 465 — 470
CHAPTER XVI.
A DIRECT METHOD OF OBTAINING THE EQUATIONS CONNECTING THETA
PRODUCTS.
290 Contents of this chapter 471
291 An addition theorem obtained by multiplying two theta functions . 471 — 474
292 An addition theorem obtained by multiplying four theta functions 474 — 477
293 The general formula obtained by multiplying any number of
theta functions . . . . . . 477 485
CHAPTER XVII.
THETA RELATIONS ASSOCIATED WITH CERTAIN GROUPS OF CHARACTERISTICS.
294 Abbreviations. Definition of syzygetic and azygetic. References
to literature (see also p. 296) 486, 487
295 A preliminary lemma . 488
296 Determination of a Gopel group of characteristics . . . 489, 490
297 Determination of a Gopel system of characteristics . . . 490, 491
298, 299 Determination and number of Gopel systems of the same parity 492 — 494
300 — 303 Determination of a fundamental set of Gopel systems . . 494 — 501
304, 305 Statement of results obtained, with the simpler applications . 502 — 504
306 — 308 Number of linearly independent theta functions of the second
order of a particular kind. Explicit mention of an import
ant identity 505 — 510
309 — 311 The most important formulae of the chapter. A general addi
tion theorem. The g> function expressed by quotients of
theta functions 510 — 516
312 — 317 Other applications of the principles of the chapter. The expres
sion of a function 3 (nv) as an integral polynomial of order
«2 in 2" functions $(v) 517—527
CONTENTS.
XV11
CHAPTER XVIII.
TRANSFORMATION OF PERIODS, ESPECIALLY LINEAR TRANSFORMATION.
§§ PAGES
318 Bearings of the theory of transformation 528, 529
319 — 323 The general theory of the modification of the period loops on a
Riemann surface 529 — 534
324 Analytical theory of transformation of periods and characteristic
of a theta function 534 — 538
325 Convergence of the transformed function . . . . . 538
326 Specialisation of the formulae, for linear transformation . . 539, 540
327 Transformation of theta characteristics ; of even characteristics ;
of syzygetic characteristics . . . . . . .541, 542
328 Period characteristics and theta characteristics . . . • . 543
329 Determination of a linear transformation to transform any even
characteristic into the zero characteristic .... 544, 545
330, 331 Linear transformation of two azygetic systems of theta charac
teristics into one another . . . . . . . . 546 — 550
332 Composition of two transformations of different orders ; supple
mentary transformations 551, 552
333, 334 Formation of p + 2 elementary linear transformations by the
composition of which every linear transformation can be
formed ; determination of the constant factors for each of
these ,.:;./.,. Lrj . 553—557
335 The constant factor for any linear transformation . . . 558, 559
336 Any linear transformation may be associated with a change of
the period loops of a Riemann surface 560, 561
337, 338 Linear transformation of the places mlt ..., mp .•* • . . . 562
339 Linear transformation of the characteristics of a radical function 563, 564
340 Determination of the places Wj, ..., mp upon a Riemann surface
whose mode of dissection is assigned . ..».'..•. .. 565 — 567
341 Linear transformation of quotients of hyperelliptic theta functions 568
342 A convenient form of the period loops in a special hyperelliptic
case. Weierstrass's number notation for half-integer charac
teristics . . . ..:;.:,;; .;. • . •..:. . . . 569, 570
CHAPTER XIX.
ON SYSTEMS OF PERIODS AND ON GENERAL JACOBIAN FUNCTIONS.
343
344—350
571
571—579
Scope of this chapter .........
Columns of periods. Exclusion of infinitesimal periods. Expres
sion of all period columns by a finite number of columns,
with integer coefficients
351 — 356 Definition of general Jacobian function, and comparison with
theta function 579 588
357—362 Expression of Jacobian function by means of theta functions.
Any p + 2 Jacobian functions of same periods and parameter
connected by a homogeneous polynomial relation . . 588 — 598
XV111 CONTENTS.
CHAPTER XX.
TRANSFORMATION OF THETA FUNCTIONS.
§§ PAGES
363 Sketch of the results obtained. References to the literature . 599, 600
364, 365 Elementary theory of transformation of second order . . . 600 — 606
366, 367 Investigation of a general formula preliminary to transformation
of odd order 607—610
368, 369 The general theorem for transformation of odd order . . . 611 — 616
370 The general treatment of transformation of the second order . 617—619
371 The two steps in the determination of the constant coefficients 619
372 The first step in the determination of the constant coefficients 619 — 622
373 Remarks and examples in regard to the second step . . . 622 — 624
374 Transformation of periods when the coefficients are not integral 624 — 628
375 Reference to the algebraical applications of the theory . . 628
CHAPTER XXI.
COMPLEX MULTIPLICATION OF THETA FUNCTIONS. CORRESPONDENCE
OF POINTS ON A RlEMANN SURFACE.
376 Scope of the chapter . . . 629
377, 378 Necessary conditions for a complex multiplication, or special
transformation, of theta functions ...... 629 — 632
379 — 382 Proof, in one case, that these conditions are sufficient . . 632 — 636
383 Example of the elliptic case 636—639
384 Meaning of an (r, s) correspondence on a Riemann surface . 639, 640
385 Equations necessary for the existence of such a correspondence 640 — 642
386 Algebraic determination of a correspondence existing on a per
fectly general Riemann surface . ... . . . . 642 — 645
387 The coincidences. Examples of the inflections and bitangents of
a plane curve 645 — 648
388 Conditions for a (1, s) correspondence on a special Riemann surface 648, 649
389 When p>l a (1, 1) correspondence is necessarily periodic . . 649, 650
390 And involves a special form of fundamental equation . . 651
391—393 When p>l there cannot be an infinite number of (1, 1) corre
spondences 652 — 654
394 Example of the case p = l 654—656
CHAPTER XXII.
DEGENERATE ABELIAN INTEGRALS.
395 Example of the property to be considered . . . . 657
396 Weierstrass's theorem. The property involves a transformation
leading to a theta function which breaks into factors . . 657, 658
CONTENTS.
XIX
397 Weierstrass's and Picard's theorem. The property involves a
linear transformation leading to T^'2 = l/r.
398 Existence of one degenerate integral involves another (p = 2)
399, 400 Connection with theory of special transformation, when p — Z .
401 — 403 Determination of necessary form of fundamental equation.
Eeferences
PAGES
658, 659
659
660, 661
661—663
404
APPENDIX I.
ON ALGEBRAIC CURVES IN SPACE.
Formal proof that an algebraic curve in space is an interpreta
tion of the relations connecting three rational functions on
a Riemann surface (cf. § 162) 664,
665
APPENDIX II.
ON MATRICES.
405 — 410 Introductory explanations . . . ; ., . , ... 666 669
411 — 415 Decomposition of an Abelian matrix into simpler ones . ' . 669 674
416 A particular result ... 674
417, 418 Lemmas , . . 675
419, 420 Proof of results assumed in §§ 396, 397 .... .'. 675, 676
INDEX OF AUTHORS QUOTED .
TABLE OF SOME FUNCTIONAL SYMBOLS
SUBJECT INDEX
677, 678
679
680—684
ADDENDA. CORRIGENDA.
PAGE LINE
6, 2, for bb^da, read (tb*~lda.
8, 22, for deficiency 1, read deficiency 0.
11, 12, for 2n-2+p, read 2n-2 + 2p.
16, § 16, 4, for called, read applied to.
dx . dx
18, 25, for — , read — .
x y
37, 31, for in, read is.
38, 3, for surfaces, read surface.
43, 20, for w, read w.
56, 22, for (x-af~\ read (x -a)P-A+1.
61, 24, add or g{ (x, y).
66, 22, for r'-l, read Tj'-l.
70, 14, for rr+l, read rr+l.
73, 28, for x'^'^''2 sl5 2, read x~2r'~2 slt j.
81. The argument of § 52 supposes p>l.
104, § 72. See also Hensel, Crelle, cxv. (1895).
114, 3 from the bottom, add here.
137. To the references, add, Macaulay, Proc. Lon. Math. Soc., xxvi. p. 495.
157. See also Kraus, Math. Annal. xvi. (1879).
166. See also Zeuthen, Ann. d. Mat. 2a Ser., t. in. (1869).
189, 21, for xii, read xi.
196, 23, for \h, read \h.
24, for \h, read \h.
197, 24, for A, read B.
198, 5, for ^(w')"1^, read y(u')~lu.
18, for fourth minus sign, read sign of equality.
206, 4, supply dz, after third integral sign: the summation is from k = 2, fc'=0.
5, supply dz, after first integral sign.
8, for $(X)l<t>(X), read 0'(*)/0(X).
247, 11. Positive means >0. The discriminant must not vanish.
6 from bottom. Cf. p. 531, notef.
282, 11, for ft, read O.
284, 18, the equation is httP = iriP + bP'.
316, 3 from the bottom, for u, read UQ.
320, heading, destroy full stop.
327, 23, for Pi(xp), read /J.J(XP).
340. Further references are given in the report of Brill and Noether (see
Preface), p. 473.
342. For various notations for characteristics see the references in the report of
Brill and Noether, p. 519.
379, 16, for T(II, ritp, read v^-", vpx'a.
420, 18, read ...characteristic, other than the zero characteristic, as the sum of two
different odd half-integer characteristics in
441, 15, for one, read in turn every combination.
533, 13. The relation had been given by Frobenius.
557, 15, for .w2, read w-?.
575, 20, for from, read for.
587, 8 and 11 ; the quantity is AeA.
In this volume no account is given of the differential equations satisfied by the theta
functions, or of their expansion in integral powers of the arguments. The following refer
ences may be useful : Wiltheiss, Crelle, xcix., Math. Annal. xxix., xxxi., xxxin., Gotting.
Nachr., 1889, p. 381; Pascal, Gotting. Xachr., 1889, pp. 416, 547, Ann. di Mat., Ser. 2% t.
xvii.; Burkhardt (and Klein), Math. Annal. xxxn. The case p — 2 is considered in Krause,
Transf. Hyperellip. Functionen.-
The following books of recent appearance, not referred to in the text, may be named here.
(1) The completion of Picard, Traite d'Analyse, (2) Jordan, Cours d'Analyse, t. n. (1894),
(3) Appell and Goursat, Theorie des Fonctions algebriques et de leurs integrates (1895), (4)
Stahl, Theorie der AbeVschen Functionen (1896).
CHAPTER I.
ADDITIONAL CORRECTIONS FOR BAKER'S ABELIAN FUNCTIONS.
PAGE LINE
138, 14, from the bottom, for greater, read less.
219, 12, 13, from the bottom, for r, read R.
315, 6, from the bottom, for f, read f.
316, 5, from the bottom, for u, read u0.
317, 4, from the bottom, for a, vanishes, &j , read, respectively, bl , is infinite, a.
333, 3, for the first + , read - .
333, 3, 7, 8, from the bottom, for A, read V.
334, 6, 7, from the bottom, for pt, pj, read p$, pf'2.
335, 12, from the bottom, for A, read \ .
340, 6, from the bottom, for Gopel, read Kummer. Supply also the reference,
Weber, Crelle LXXXIV. (1878), p. 341.
359, 1, after periods, add and let ^ (u) = @ (u) + @(u + u').
5, for $>, read ^ ; for iir, read Ziir.
9, for P + iQ, read (P + iQ) [(£>' (u) + $' (v)], where u, v are the arguments occurring
in the denominator ; and similarly for P-iQ ; and add to the function
the term 4P f (u) - — , f (w')l, where f(w) is Weierstrass's function.
367, 5, from the bottom, for m, read p.
444, 16, for x, read u.
445, 14, for n, read p.
457, 14, from the bottom, supply the reference, § 181.
615, 5, for xviii., read xvn.
665, 6, from the bottom, add, which may be taken to be linear polynomials in
x only.
sheet. Or the sheets may wind into one another : in which case we shall
regard this winding point (or branch point) as constituting one place : this
place belongs then indifferently to either sheet ; the sheets here merge into
one another. In the first case, if a be the value of x for which the sheets
just touch, supposed for convenience of statement to be finite, and x a value
* For references see Chap. II. § 12, note.
t Such a point is called by Riemann "ein sich aufhebender Verzweigungspunkt " : Gesam-
melte Werke (1876), p. 105.
B. 1
ADDENDA. CORRIGENDA.
PAGE LINE
^
6, 2, for db^da, read db^~da.
8, 22, for deficiency 1, read deficiency 0.
11, 12, for 2n-2+p, read 2n-2 + 2p.
16, § 16, 4, for called, read applied to.
dx , da;
18, 25, for — , read — .
x y
37, 31, for in, read is.
38, 3, for surfaces, read surface.
43, 20, for w, read w.
56, 22, for (x-af~\ read (x-a)<>-*+1.
fil 24. add or n, (x. u\.
587, 8 and 11 ; the quantity is AeA.
In this volume no account is given of the differential equations satisfied by the theta
functions, or of their expansion in integral powers of the arguments. The following refer
ences may be useful : Wiltheiss, Crelle, xcix., Hath. Annal. xxix., xxxi., xxxm., Gotting.
Nachr., 1889, p. 381; Pascal, Gotting. Nachr., 1889, pp. 416, 547, Ann. di Mat., Ser. 2% t.
xvii.; Burkhardt (and Klein), Math. Annal. xxxn. The case p = 2 is considered in Krause,
Transf. Hyperellip. Functionen.-
The following books of recent appearance, not referred to in the text, may be named here.
(1) The completion of Picard, Traite d'Anatyse, (2) Jordan, Cours d'Analyse, t. 11. (1894),
(3) Appell and Goursat, Theorie des Fonctions algebriques et de leurs integrates (1895), (4)
Stahl, Theorie der AbeVschen Functionen (1896).
CHAPTER I.
THE SUBJECT OF INVESTIGATION.
1. THIS book is concerned with a particular development of the theory
of the algebraic irrationality arising when a quantity y is defined in terms
of a quantity x by means of an equation of the form
a0yn + atf1'1 +...+ an^y + an = 0,
wherein a0, al} ...,an are rational integral polynomials in x. The equation is
supposed to be irreducible ; that is, the left-hand side cannot be written as
the product of other expressions of the same rational form.
2. Of the various means by which this dependence may be represented,
that invented by Riemann, the so-called Riemann surface, is throughout
regarded as fundamental. Of this it is not necessary to give an account
here*. But the sense in which we speak of a place of a Riemann surface
must be explained. To a value of the independent variable x there will in
general correspond n distinct values of the dependent variable y — represented
by as many places, lying in distinct sheets of the surface. For some values
of x two of these n values of y may happen to be equal : in that case the
corresponding sheets of the surface may behave in one of two ways. Either
they may just touch at one point without having any further connexion in
the immediate neighbourhood of the point t : in which case we shall regard
the point where the sheets touch as constituting two places, one in each
sheet. Or the sheets may wind into one another : in which case we shall
regard this winding point (or branch point) as constituting one place : this
place belongs then indifferently to either sheet ; the sheets here merge into
one another. In the first case, if a be the value of x for which the sheets
just touch, supposed for convenience of statement to be finite, and x a value
* For references see Chap. II. § 12, note.
t Such a point is called by Riemann "ein sich aufhebender Verzweigungspunkt " : Gesam-
melte Werke (1876), p. 105.
B. 1
2 THE PLACES OF A RIEMANN SURFACE. [2
very near to a, and if b be the value of y at each of the two places, also
supposed finite, and ylt yz be values of y very near to b, represented by
points in the two sheets very near to the point of contact of the two
sheets, each of 3/1 — 6, yz — b can be expressed as a power-series in x — a
with integral exponents. In the second case with a similar notation each
of 2/1 — 6, y2 — 6 can be expressed as a power-series in (x — a)* with integral
exponents. In the first case a small closed curve can be drawn on either
of the two sheets considered, to enclose the point at which the sheets touch :
and the value of the integral •= — . Id log (x - a) taken round this closed curve
will be 1 ; hence, adopting a definition given by Riemann*, we shall say that
x — a is an infinitesimal of the first order at each of the places. In the
second case the attempt to enclose the place by a curve leads to a curve
lying partly in one sheet and partly in the other; in fact, in order that
the curve may be closed it must pass twice round the branch place. In this
case the integral ^ — . Id log [(x — a)*] taken round the closed curve will be 1 :
and we speak of (x — a}*- as an infinitesimal of the first order at the place.
In either case, if t denote the infinitesimal, x and y are uniform functions
of t in the immediate neighbourhood of the place ; conversely, to each point
on the surface in the immediate neighbourhood of the place there corre
sponds uniformly a certain value of if. The quantity t effects therefore a
conformal representation of this neighbourhood upon a small simple area in
the plane of t, surrounding t — 0.
3. This description of a simple case will make the general case clear.
In general for any finite value of x, x = a, there may be several, say k, branch
points J; the number of sheets that wind at these branch points may be
denoted by w1+l,w.2+l, . .., wk+ 1 respectively, where
(w1 + 1) + (w, + l) + ...+(wk+l) = n,
so that the case of no branch point is characterised by a zero value of the
corresponding w. For instance in the first case above, notwithstanding that
two of the n values of y are the same, each of w1} w.2, ...,Wk is zero and k is
equal to n : and in the second case above, the values are k = n — 1, wr = 1, w.2 = 0,
w3 = 0, . . . , wk = 0. In the general case each of these k branch points is called a
place, and at these respective places the quantities (x - a)w>+l, ..., (x— a)wt+l
* Gesammelte Werke (1876), p. 96.
+ The limitation to the immediate neighbourhood involves that t is not necessarily a rational
function of x, y.
It may be remarked that a rational function of x and y can be found whose behaviour in
the neighbourhood of the place is the same as that of t. See for example Hamburger,
Zeitschrift f. Math, und Phys. Bd. 16, 1871 ; Stolz, Math. Ann. 8, 1874 ; Harkness and Morley,
Theory of Functions, p. 141.
t Cf. Forsyth, Theory of Functions, p. 171. Prym, Crelle, Bd. 70.
4] TRANSFORMATION OF THE EQUATION. 3
are infinitesimals of the first order. For the infinite value of x we shall
similarly have n or a less number of places and as many infinitesimals, say
-_
+1, ..., (-r'+1, where (Wl + l)+ ... +(w,. + I) = n. And as in the par-
xj \x/
ticular cases discussed above, the infinitesimal t thus defined for every place
of the surface has the two characteristics that for the immediate neighbour
hood of the place x and y are uniquely expressible thereby (in series of
integral powers), and conversely t is a uniform function of position on the
surface in this neighbourhood. Both these are expressed by saying that
t effects a reversible conformal representation of this neighbourhood upon a
simple area enclosing t = 0. It is obvious of course that quantities other
than t have the same property.
A place of the Riemann surface will generally be denoted by a single
letter. And in fact a place (x, y} will generally be called the place x.
When we have occasion to speak of the (n or less) places where the inde
pendent variable x has the same value, a different notation will be used.
4. We have said that the subject of enquiry in this book is a certain
algebraic irrationality. We may expect therefore that the theory is practi
cally unaltered by a rational transformation of the variables x, y which is of
a reversible character. Without entering here into the theory of such trans
formations, which comes more properly later, in connexion with the theory
of correspondence, it is necessary to give sufficient explanations to make it
clear that the functions to be considered belong to a whole class of Riemann
surfaces and are not the exclusive outcome of that one which we adopt initially.
Let £ be any one of those uniform functions of position on the funda
mental (undissected) Riemann surface whose infinities are all of finite order.
Such functions can be expressed rationally by x and y*. For that reason we
shall speak of them shortly as the rational functions of the surface. The
order of infinity of such a function at any place of the surface where the
function becomes infinite is the same as that of a certain integral power of
the inverse - of the infinitesimal at that place. The sum of these orders of
6
infinity for all the infinities of the function is called the order of the function.
The number of places at which the function f assumes any other value a is
the same as this order : it being understood that a place at which £ — a is
zero in a finite ratio to the rth order of t is counted as r places at which £ is
equal to off. Let v be the order of £. Let T? be another rational function of
* Forsyth, Theory of Functions, p. 370.
t For the integral — /dlog(£-a), taken round an infinity of log(£-a), is equal to the
order of zero of £ - a at the place, or to the negative of the order of infinity of £, as the case may
be. And the sum of the integrals for all such places is equal to the value round the boundary of
the surface— which is zero. Cf. Forsyth, Theonj of Functions, p. 372.
1—2
4 CONDITION OF REVERSIBILITY. [4
order p. Take a plane whose real points represent all the possible values of
|f in the ordinary way. To any value of |f, say |f = a, will correspond v
positions Xlt ..., Xvon the original Riemann surface, those namely where £
is equal to a : it is quite possible that they lie at less than v places of the
surface. The values of 77 at X1} ..., Xv may or may not be different. Let
H denote any definite rational symmetrical function of these v values of 77.
Then to each position of a in the |f plane will correspond a perfectly unique
value of H, namely, H is a one-valued function of £. Moreover, since 77 and
|f are rational functions on the original surface, the character of H for values
of |f in the immediate neighbourhood of a value a, for which H is infinite, is
clearly the same as that of a finite power of ff — a. Hence H is a rational
function of |f. Hence, if Hr denote the sum of the products of the values of
i] at Xlt ..., Xv, r together, 77 satisfies an equation
r)"-r)"^H1 + r)^H2-...+(-YHv = 0>
whose coefficients are rational functions of |f.
It is conceivable that the left side of this equation can be written as the
product of several factors each rational in |f and 77. If possible let this be
done. Construct over the |f plane the Riemann surfaces corresponding to
these irreducible factors, 77 being the dependent variable and the various
surfaces lying above one another in some order. It is a known fact, already
used in defining the order of a rational function on a Riemann surface, that
the values of 77 represented by any one of these superimposed surfaces in
clude all possible values — each value in fact occurring the same number of
times on each surface. To any place of the original surface, where |f, 77 have
definite values, and to the neighbourhood of this place, will correspond there
fore a definite place (|f, 77) (and its neighbourhood) on each of these super
imposed surfaces. Let 77!, ...,tjr be the values of 77 belonging, on one of
these surfaces, to a value of £ : and T?/, ..., r}s' the values belonging to the
same value of |f on another of these surfaces. Since for each of these surfaces
there are only a finite number of values of £ at which the values of 77 are
not all different, we may suppose that all these r values on the one
surface are different from one another, and likewise the s values on the other
surface. Since each of the pairs of values (|f, 77^, . . . , (|f, r)r) must arise on
both these surfaces, it follows that the values 77!, ...,tjr are included among
77/, ..., 77/. Similarly the values T7/, ..., i?/ are included among 77^ ...,77,..
Hence these two sets are the same and r = s. Since this is true for an
infinite number of values of |f, it follows that these two surfaces are merely
repetitions of one another. The same is true for every such two surfaces.
Hence r is a divisor of v and the equation
when reducible, is the v/rih power of a rational equation of order r in 77. It
will be sufficient to confine our attention to one of the factors and the (£, 77)
5] CORRESPONDENCE OF TWO SURFACES. 5
surface represented thereby. Let now Xlt . . . , Xv be the places on the original
surface where £ has a certain value. Then the values of 77 at Xlt . . , Xv will
consist of v/r repetitions of r values, these r values being different from one
another except for a finite number of values of £ Thus to any place (f, 77) on
one of the v/r derived surfaces will correspond v/r places on the original
surface, those namely where the pair (£, 77) take the supposed values. Denote
these by PlfPa, — Let Y be any rational symmetrical function of the v/r
pairs of values (a}1} y^), (#2, 2/2)» •••> which the fundamental variables a, y of the
original surface assume at P1; P2) — Then to any pair of values (£, 77) will
correspond only one value of Y — namely, Y is a one-valued function on the
(£, 77) surface. It has clearly also only finite orders of infinity. Hence Y is
a rational function of £, 77. In particular #u #2, ... are the roots of an
equation whose coefficients are rational in £, 77 — as also are yi} yz, ____
There exists therefore a correspondence between the (£, 77) and (x, y)
surfaces — of the kind which we call a (1, - j correspondence: to every place
of the (x, y) surface corresponds one place of the (£, 77) surface; to every
place of this surface correspond - places of the (x, y) surface.
The case which most commonly arises is that in which the rational
irreducible equation satisfied by 77 is of the vih degree in 77: then only one
place of the original surface is associated with any place of the new surface.
In that case, as will appear, the new surface is as general as the original
surface. Many advantages may be expected to accrue from the utilization of
that fact. We may compare the case of the reduction of the general equation
of a conic to an equation referred to the principal axes of the conic.
5. The following method* is theoretically effective for the expression of x, y in terms
of & r,.
Let the rational expression of £, rj in terms of x, y be given by
<£ (x, y) - & (x, y ) = 0, ^ (x, y} - rfX (x, y} = 0,
and let the rational result of eliminating #, y between these equations and the initial
equation connecting x, y be denoted by F(£, rj) = 0, each of $, ..., ^, ^denoting integral
polynomials. Let two terms of the expression (f>(z, y) — ty(&, y) = 0 be axry*—t-bxr'y*'.
This expression and therefore all others involved will be unaltered if «, 6 be replaced by
such quantities a + h, b + k, that hxry*=z£kxr'y*'. In a formal sense this changes F(£, rj)
into
where X ^ 1, and F is such that all differential coefficients of it in regard to a and b of order
less than X are identically zero.
Hence the term within the square brackets in this expression must be zero. If it is
possible, choose now r = rf + \ and s = s', so that k=
* Salmon's Higher Algebra (1885), p. 97, § 103.
ALGEBRAICAL FORMULATION. [5
Then we obtain the equation
This is an equation of the form above referred to, by which x is determinate from £ and
T]. And y is similarly determinate.
It will be noticed that the rational expression of xt y by £, rj, when it is possible
from the equations
will not be possible, in general, from the first two equations : it is only the places x, y
satisfying the equation f(x, y) = Q which are rationally obtainable from the places £, 17
satisfying the equation F(£, r)) = 0. There do exist transformations, rationally reversible,
subject to no such restriction. They are those known as Cremona-transformations*.
They can be compounded by reapplication of the transformation x : y : I = rj : {• : £»/.
We may give an example of both of these transformations —
For the surface
the function £=y2/(^2 + .£ + l) is of order 2, being infinite at the places where x2+z+l = 0,
in each case like (x-a)~°, and the function r}=x/y is of order 4, being infinite at the
places x*+x+I=0, in each case like (^-a)"t, a being the value of x at the place.
From the given equation we immediately find, as the relation connecting £ and 17,
and infer, since the equation formed as in the general statement above should be of
order 2 in rj, that this general equation will be
Thence in accordance with that general statement we infer that to each place (£, >;) on
the new surface should correspond two places of the original surface : and in fact these are
obviously given by the equations
r}^=^/
If however we take
£=y2/(#2
where « is an imaginary cube root of unity, so that 17 is a function of order 3, these
equations are reversible independently of the original equation, giving in fact
x = („£ _ wy )/(£ - ^}, y = (m- 2
and we obtain the surface
having a (1, 1) correspondence with the original one.
It ought however to be remarked that it is generally possible to obtain reversible
transformations which are not Cremona-transformations.
6. When a surface (x, y) is (1,1) related to a (£, 77) surface, the defi
ciencies of the surfaces, as denned by Riemann by means of the connectivity,
must clearly be the same.
* See Salmon, Higher Plane Curves (1879), § 362, p. 322.
6] RELATION OF DEF1CIENCES. 7
It is instructive to verify this from another point of view*. — Consider at
how many places on the original surface the function -~ is zero. It is infinite
CLOG
at the places where % is infinite: suppose for simplicity that these are
separated places on the original surface or in other words are infinities of
the first order, and are not at the branch points of the original surface. At
d£ 1
a pole of £, ,- is infinite twice. It is infinite like — at a branch place (a)
CLOG v
where x — a = tw+l: namely it is infinite ^w = 2n + '2p - 2 times t at the branch
places of the original surface. It is zero 2n times at the infinite places of the
original surface. There remain therefore 2v + 2n + 2p — 2 — 2n = 2v + 2p — 2
places where ~ is zero. If a branch place of the original surface be a pole
1 -7fc 1
of £, and £ be there infinite like -, -~ is infinite like - — — , namely 2+w
t ax t2 . tw
times : the total number of infinities of -^ will therefore be the same as
dx
7«-
before. Now at a finite place of the original surface where -r = 0, there are
ax
two consecutive places for which £ has the same value. Since - = 1 they can
only arise from consecutive places of the new surface for which £ has the
same value. The only consecutive places of a surface for which this is the
case are the branch places. Hence f there are 2v+2p — 2 branch places of
the new surface. This shews that the new surface is of deficiency p.
When v/r is not equal to 1, the case is different. The consecutive places
of the old surface, for which £ has the same value, may either be those arising
from consecutive places of the new surface — or may be what we may call
accidental coincidences among the v/r places which correspond to one place
of the new surface. Conversely, to a branch place of the new surface,
characterised by the same value for £ for consecutive placesj, will correspond
vjr places on the old surface where £ has the same value for consecutive
places. In fact to two very near places of the new surface will correspond
v/r pairs each of very near places on the old surface. If then C denote the
number of places on the old surface at which two of the v/r places corre
sponding to a place on the new surface happen to coincide, and w' the number
of branch points of the new surface, we have the equation
'-
r
* Compare the interesting geometrical account, Salmon, Higher Plane Curves (1879), p. 326,
§ 364, and the references there given.
t Forsyth, Tlieory of Functions, p. 348.
:£ Namely, near such a branch place f = a, £ - a is zero of higher order than the first.
8 PARAMETERS NOT REMOVED [6
and if p be the deficiency of the new surface (of r sheets), this leads to the
equation
f
(2r + 2pf
from which
Corollary*. If p =p', then C = (2p - 2) (l - -\ . Thus - > 1, so that
(7 = 0, and the correspondence is reversible.
We have, herein, excluded the case when some of the poles of £ are of
higher than the first order. In that case the new surface has branch places
at infinity. The number of finite branch places is correspondingly less. The
reader can verify that the general result is unaffected.
Ex. In the example previously given (§ 5) shew that the function £ takes any given
value at two points of the original surface (other than the branch places where it is
infinite), 17 having the same value for these two points, and that there are six places at
which these two places coincide. (These are the place (# = 0, y = 0) and the five places
where x= — 2.)
There is one remark of considerable importance which follows from the
theory here given. We have shewn that the number of places of the (x, y)
surface which correspond to one place of the (£, 97) surface is - , where v is the
order of £ and r is not greater than v, being the number of sheets of the (f , 77)
surface ; hence, if there were a function £ of order 1 the correspondence would
be reversible and therefore the original surface would be of deficiency 1.
7. This notion of the transformation of a Riemann surface suggests an
inference of a fundamental character.
The original equation contains only a finite number of terms : the original
surface depends therefore upon a finite number of constants, namely, the
coefficients in the equation. But conversely it is not necessary, in order that
the equation be reversibly transformable into another given one, that the
equation of the new surface contain as many constants as that of the original
surface. For we may hope to be able to choose a transformation whose
coefficients so depend on the coefficients of the original equation as to reduce
this number. If we speak of all surfaces of which any two are connected by
a rational reversible transformation as belonging to the same class f, it becomes
a question whether there is any limit to the reduction obtainable, by rational
reversible transformation, in the number of constants in the equation of a
surface of the class.
* See Weber, Crelle, 76, 345.
t So that surfaces of the same class will be of the same deficiency.
7] BY TRANSFORMATION. 9
It will appear in the course of the book* that there is a limit, and that
the various classes of surfaces of given deficiency are of essentially different
character according to the least number of constants upon which they depend.
Further it will appear, that the most general class of deficiency p is
characterised by 3p — 3 constants when p > 1 — the number for p = 1 being
one, and for p = 0 none.
For the explanatory purposes of the present Chapter we shall content
ourselves with the proof of the following statement — When a surface is
reversibly transformed as explained in this Chapter, we cannot, even though
we choose the new independent variable £ to contain a very large number of
disposeable constants, prescribe the position of all the branch points of the
new surface ; there will be 3p — 3 of them whose position is settled by the
position of the others. Since the correspondence is reversible we may regard
the new surface as fundamental, equally with the original surface. We
infer therefore that the original surface depends on 3p — 3 parameters —
or on less, for the 3/> — 3 undetermined branch points of the new surface may
have mutually dependent positions.
In order to prove this statement we recall the fact that a function
of order Q contains^ Q—p + l linearly entering constants when its poles
are prescribed: it may contain more for values of Q<2p — 1, but we
shall not thereby obtain as many constants as if we suppose Q > 2p — 2
and large enough. Also the Q infinities are at our disposal. We can then
presumably dispose of 2Q-p + 1 of the branch points of the new surface.
But these are, in number, 2Q + 2p — 2 when the correspondence is reversible.
Hence we can dispose of all but 2Q + 2p - 2 - (2Q -p + 1) = 3p - 3 of the
branch points of the new surface J.
Ex. 1. The surface associated with the equation
y*=x(l -x] (l-tfx) (1 -XV) (1 -MV) (l-v*x) (1 -p%)
is of deficiency 3. It depends on 5 = 2p- 1 parameters, /c2, X2, /u2, v2, p2.
Ex. 2. The surface associated with the equation
y*+y*(x, l\+y(x, !), + (#, 1)4=0,
wherein the coefficients are integral polynomials of the orders specified by the suffixes, is
of deficiency 3. Shew that it can be transformed to a form containing only 5 = 2^-1
parametric constants.
* See the Chapters on the geometrical theory and on the inversion of Abelian Integrals. The
reason for the exception in case ^ = 0 or 1 will appear most clearly in the Chapter on the self-
correspondence of a Riemann surface. But it is a familiar fact that the elliptic functions which
can be constructed for a surface of deficiency 1 depend upon one parameter, commonly called
the modulus : and the trigonometrical functions involve no such parameter.
t Forsyth, p. 459. The theorems here quoted are considered in detail in Chapter III. of the
present book.
£ Cf. Kiemann, Ges. Werke (1876), p. 113. Klein, Ueber Riemann's Theorie (Leipzig,
Teubner, 1882), p. 65.
c;
UN I VI.
Of ~ >-.
10 SELF-CORRESPONDENCE. [8
8. But there is a case in which this argument fails. If it be possible to
transform the original surface into itself by a rational reversible transforma
tion involving r parameters, any r places on the surface are effectively
equivalent with, as being transformable into, any other r places. Then the
Q poles of the function £ do not effectively supply Q but only Q — r dispose-
able constants with which to fix the new surface. So that there are 3/> — 3 + r
branch points of the new surface which remain beyond our control. In this
case we may say that all the surfaces of the class contain 3p - 3 disposeable
parameters beside r parameters which remain indeterminate and serve to
represent the possibility of the self-transformation of the surface. It will be
shewn in the chapter on self- transformation that the possibility only arises
for p = 0 or p = 1, and that the values of r are, in these cases, respectively
3 and 1. We remark as to the case p = 0 that when the fundamental
surface has only one sheet it can clearly be transformed into itself by
a transformation involving three constants x— 5 , : and in regard to p = 1,
c% -f d
the case of elliptic functions, that effectively a point represented by the
elliptic argument u is equivalent to any other point represented by an
argument u + 7. For instance a function of two poles is
and clearly Fa>ft has the same value at u as has Fa+y>p+y at u -f 7 : so that the
poles (a, ft) are not, so far as absolute determinations are concerned, effective
for the determination of more than one point.
9. The fundamental equation
a0yn + aiyn-l + ...+an = 0,
so far considered as associated with a Riemann surface, may also be regarded
as the equation of a plane curve : and it is possible to base our theory on the
geometrical notions thus suggested. Without doing this we shall in the
following pages make frequent use of them for purposes of illustration. It is
therefore proper to remind the reader of some fundamental properties*.
The branch points of the surface correspond to those points of the curve
where a line x = constant meets the curve in two or more consecutive points :
as for instance when it touches the curve, or passes through a cusp. On the
other hand a double point of the curve corresponds to a point on the surface
where two sheets just touch without further connexion. Thus the branch
place of the surface which corresponds to a cusp is really a different singu
larity to that which corresponds to a place where the curve is touched by a
* Cf. Forsyth, Theory of Functions, p. 355 etc. Harkness and Morley, Theory of Functions,
p. 273 etc.
9] GEOMETRICAL VIEW. 11
line x = constant, being obtained by the coincidence of an ordinary branch
place with such a place of the Riemann surface as corresponds to a double
point of the curve.
Properties of either the Riemann surface or a plane curve are, in the
simpler cases, immediately transformed. For instance, by Pliicker's formulae
for a curve, since the number of tangents from any point is
f-(n-l)n-2£-3/c,
where n is the aggregate order in a; and y, it follows that the number of
branch places of the corresponding surface is
w = t + K = (n - 1) n - 2 (8 + K)
= 2n-2 + 2{iO-l)O-2)-S-4
Thus since w = 2n — 2 -j^p, the deficiency of the surface is
£0-1)0- 2)- S-K,
namely the number which is ordinarily called the deficiency of the curve.
To the theory of the birational transformation of the surface corresponds
a theory of the birational transformation of plane curves. For example, the
branch places of the new surface obtained from the surface f(x, y) = 0 by
means of equations of the form <£ (x, y} — ty (x, y) = 0, $ (x, y) — 77% (x, ?/) = 0
will arise for those values of £ for which the curve </> (x, y) — jfy (x, y) — 0
touches f(x, y} = 0. The condition this should be so, called the tact inva
riant, is known to involve the coefficients of <f> (as, y) — % \Jr (x, y~) = 0, and
therefore in particular to involve £, to a degree* n (n — 3) — 28 — 3/c + 2nn,
where n' is the order of <£ (x, y) — £i/r (x, y} = 0. Branch places of the new
surface also arise corresponding to the cusps of the original curve. The total
number is therefore n (n — 3) — 25 — 2* + Znri = *2p — 2'+ 2nn'. Now nri is
the number of intersections of the curves f(x, y) = Q and <£ (x, y) — jfy (x, y) = 0,
namely it is the number of values of t] arising for any value of £, and is
thus the number of sheets of the new surface, which we have previously
denoted by v : so that the result is as before.
In these remarks we have assumed that the dependent variable occurs
to the order which is the highest aggregate order in x and y together — and
we have spoken of this as the order of the curve. And in regarding two
curves as intersecting in a number of points equal to the product of their
orders we have allowed count of branches of the curve which are entirely
at infinity. Some care is necessary in this regard. In speaking of the
Riemann surface represented by a given equation it is intended, unless the
contrary be stated, that such infinite branches are unrepresented. As an
example the curve y- = (x, 1)6 may be cited.
Ex, Prove that if from any point of a curve, ordinary or multiple, or from a point not
on the curve, t be the number of tangents which can be drawn other than those touching
* See Salmon, Higher Plane Curves (1879), p. 81.
12 GENERALITY [9
at the point, and K be the number of cusps of the curve — and if v be the number of
points other than the point itself in which the curve is intersected by an arbitrary line
through the point— -then t + K — 2i/ is independent of the position of the point. If the
equation of the variable lines through the point be written u — gv = 0, interpret the result
by regarding the curve as giving rise to a Riemann surface whose independent variable
fa |*.
10. The geometrical considerations here referred to may however be
stated with advantage in a very general manner.
In space of any (k) dimensions let there be a curve — (a one-dimension
ality). Let points on this curve be given by the ratios of the k + 1 homo
geneous variables xly ... , xk+1. Let u, v be any two rational integral homo
geneous functions of these variables of the same order. The locus u — gv = 0
will intersect the curve in a certain number, say v, points — we assume the
curve to be such that this is the same for all values of £, and is finite. Let all
the possible values of £ be represented by the real points of an infinite plane
in the ordinary way. Let w, t be any two other integral functions of the
w
coordinates of the same order. The values of t] = — at the points where
t
u — %v = 0 cuts the curve for any specified value of £ will be v in number.
As before it follows thence that 77 satisfies an algebraic equation of order v
whose coefficients are one- valued functions of £. Since 77 can only be infinite
to a finite order it follows that these coefficients are rational functions of f .
Thence we can construct a Riemann surface, associated with this algebraic
equation connecting f and 77, such that every point of the curve gives rise to
a place of the surface. In all cases in which the converse is true we may
regard the curve as a representation of the surface, or conversely.
Thus such curves in space are divisible into sets according to their
deficiency. And in connexion with such curves we can construct all the
functions with which we deal upon a Riemann surface.
Of these principles sufficient account will be given below (Chapter VI.) :
familiar examples are the space cubic, of deficiency zero, and the most general
space quartic of deficiency 1 which is representable by elliptic functions.
11. In this chapter we have spoken primarily of the algebraic equation
— and of the curve or the Riemann surface as determined thereby. But this
is by no means the necessary order. If the Riemann surface be given, the
algebraic equation can be determined from it — and in many forms, according
to the function selected as dependent variable (y). It is necessary to keep
this in view in order fully to appreciate the generality of Riemann's methods.
For instance, we may start with a surface in space whose shape is that of an
* The reader who desires to study the geometrical theory referred to may consult : —
Cayley, Quart. Journal, vn. ; H. J. S. Smith, Proc. Lond. Math. Soc. vi. ; Noether, Math. Annul.
9 ; Brill, Math. Annal. 16 ; Brill u. Noether, Math. Annul. 7.
11] OF THE THEORY. 13
anchor ring*, and construct upon this surface a set of elliptic functions. Or
we may start with the surface on a plane which is exterior to two circles
drawn upon the plane, and construct for this surface a set of elliptic functions.
Much light is thrown upon the functions occurring in the theory by thus
considering them in terms of what are in fact different independent variables.
And further gain arises by going a step further. The infinite plane upon
which uniform functions of a single variable are represented may be regarded
as an infinite sphere ; and such surfaces as that of which the anchor ring
above is an example may be regarded as generalizations of that simple case.
Now we can treat of branches of a multiform function without the use of a
Riemann surface, by supposing the branch points of the function marked on
a single infinite plane and suitably connected by barriers, or cuts, across which
the independent variable is supposed not to pass. In the same way, for any
general Riemann surface, we may consider branches of functions which are
not uniform upon that surface, the branches being separated by drawing
barriers upon the surface. The properties obtained will obviously generalize
the properties of the functions which are uniform upon the surface.
* Forsyth, p. 318 ; Kiemann, Ges. Werke (1876), pp. 89, 415.
[12
CHAPTER II.
THE FUNDAMENTAL FUNCTIONS ON A RTEMANN SURFACE.
12. IN the present chapter the theory of the fundamental functions is
based upon certain a priori existence theorems*, originally given by
Riemann. At least two other methods might be followed : in Chapters IV.
and VI. sufficient indications are given to enable the reader to establish
the theory independently upon purely algebraical considerations : from
Chapter VI. it will be seen that still another basis is found in a preliminary
theory of plane curves. In both these cases the ideas primarily involved are
of a very elementary character. Nevertheless it appears that Riemann's
descriptive theory is of more than equal power with any other ; and that
it offers a generality of conception to which no other theory can lay claim.
It is therefore regarded as fundamental throughout the book.
It is assumed that the Theory of Functions of Forsyth will be accessible
to readers of the present book ; the aim in the present chapter has been to
exclude all matter already contained there. References are given also to
the treatise of Harkness and Morley*.
13. Let t be the infinitesimal f at any place of a Riemann surface : if it is
a finite place, namely, a place at which the independent variable x is finite,
the values of x for all points in the immediate neighbourhood of the place
are expressible in the form x = a + tw+1 : if an infinite place, x = t~(w+1>.
There exists a function which save for certain additive moduli is one-valued
on the whole surface and everywhere finite and continuous, save at the
place in question, in the neighbourhood of which it can be expressed in the
form
* See for instance : Forsyth, Theory of Functions of a Complex Variable, 1893 ; Harkness and
Morley, Treatise on the Theory of Functions, 1893 ; Schwarz, Gesam. math. Abhandlungen, 1890.
The best of the early systematic expositions of many of the ideas involved is found in
C. Neumann, Vorlesungen ilber Riemann's Theorie, 1884, which the reader is recommended to
study. See also Picard, Traite d" Analyse, Tom. n. pp. 273, 42 and 77.
t For the notation see Chapter I. §§ 2, 3.
14] ELEMENTARY NORMAL INTEGRALS. 15
Herein, as throughout, P (t) denotes a series of positive integral powers of t
vanishing when t = 0, G, A, ... , Ar^, are constants whose values can be
arbitrarily assigned beforehand, and r is a positive integer whose value can be
assigned beforehand.
We shall speak of all such functions as integrals of the second kind :
but the name will be generally restricted to that * particular function whose
behaviour near the place is that of
This function is not entirely unique. We suppose the surface dissected
by 2p cutsf, which we shall call period loops; they subserve the purpose of
rendering the function one-valued over the whole of the dissected surface.
We impose the further condition that the periods of the function for transit
across the p loops of the first kind j shall be zero ; then the function is unique
save for an additive constant. It can therefore be made to vanish at an
arbitrary place. The special function§ so obtained whose infinity is that
of - - is then denoted by Tax> c, c denoting the place where the function
vanishes and as the current place. When the infinity is an ordinary place,
at which either sc = a or # = oo , the function is infinite either like ----
x — a
or - x. The periods of T/' * for transit of the period loops of the second
kind will be denoted by fl1} ..., flp.
14. Let Oi^/i), (#ay2) be any two places of the surface: and let the
infinitesimals be respectively denoted by tlt L, so that in the neighbourhood
of these places we have the equations x — xl = £1W]+1, ac — x2 = t.?'*+1. Let a
cut be made between the places (a?,^), (#2<y2). There exists a function, here
denoted by n*1 c , which (a) is one-valued over the whole dissected surface,
3-1, <<2
(/3) has p periods arising for transit of the period loops of the second kind
and has no periods at the period loop of the first kind, (7) is everywhere
continuous and finite save near (a^) and (x.,ij.^), where it is infinite re
spectively like log£j and -logt,, and, (8), vanishes when the current place
denoted by x is the place denoted by c. This function is unique. If the
cut between (a?^), (aray2) be not made, the function is only definite apart
from an additive integral multiple of 2iri, whose value depends on the
* This particular function is also called an elementary integral of the second kind.
t Those ordinarily called the a, b curves; see Forsyth, p. 354. Harkness and Morley,
p. 242, etc.
£ Those called the a cuts. ^,-
§ The fact that the function has no periods at the period loops of the first kind is gene
rally denoted by calling the function a normal integral of the second kind.
16
ELEMENTARY NORMAL INTEGRALS.
[14
path by which the variable is supposed to pass from c. It will be called* the
integral of the third kind whose infinity is like that of Iog(tift2).
15. Beside these functions there exist also certain integrals of the first
kind — in number p. They are everywhere continuous and finite and one-
valued on the dissected surface. For transit of the period loops of the
first kind, one of them, say Vi, has no periods except for transit of the iih loop,
ai. This period is here taken to be 1. The periods of Vi for transit of the
period loops of the second kind are here denoted by rtV ..., T;P. We may
therefore form the scheme of periods
a.
do
dp
frl
k
•Si
1
0
0
TU
T1P
v.2
0
1
0
T21
T2P
•
VP
0
0
1
Tfl
Tpp
Each of these functions v^ is unique when a zero is given. They will there
fore be denoted by v*' °, ..., vpx> c, the zero denoted by c being at our disposal.
The periods ry- have certain properties which will be referred to in their
proper place : in particular ry- = T^, so that they are certainly not equivalent
to more than %p (p + 1) algebraically independent constants. As a fact, in
accordance with the previous chapter, when p > 1 they are subject to
l)- (3p - 3) = %(p - 2) (p - 3) relations.
16. In regard to these enunciations, the reader will notice that the word
period here used for that additive constant arising for transit of a period loop
— namely, in consequence of a path leading from one edge of the period loop
to the opposite edge — would be more properly called the period for circuit of
this path than the period for transit of the loop.
The integrals here specified are more precisely called the normal ele
mentary integrals of their kinds. The general integral of the first kind is a
linear function of Vj , . . . , vp with constant coefficients ; its periods at the first
p loops will not have the same simple forms as have those of ^ ... vp. The
general integral of the third kind, infinite like C log (t^/t^, G being a constant,
is obtained by adding a general integral of the first kind to CHJ x ; similarly
for the general integral of the second kind.
The function II*' ° hasf the property expressed by the equation
X, C
* More precisely, the normal elementary integral of the third kind,
t Forsyth, p. 453. Harkness and Morley, p. 445.
16] VARYING PARAMETER NEARLY EQUAL TO ARGUMENT. 17
A more general integral of the third kind having the same property is
wherein the arbitrary coefficients satisfy the equations Ay = Aji. The pro
perty is usually referred to as the theorem of the interchange of argument
(a1) and parameter (a^).
The property allows the consideration of
Il
*1 , 2
as a function of x^ for fixed positions of x, c, x». In this regard a remark
should be made :
For an ordinary position of x, the function
is a finite continuous function of ar/ when #/ is in the neighbourhood of x.
But if xl be a branch place where w+l sheets wind, and #/, x be two
positions in its neighbourhood, the functions of x
IT,' -log (a?/-*), Ux'c -- — 1log(a; ,-x)
*1 , *2 *„ X2 W+l
are respectively finite as x approaches #/ and aclt so that
is not a finite and continuous function of x/ for positions of a-/ up to and
including the branch place a?lt
In this case, let the neighbourhood of the branch place be conformally
represented upon a simple plane closed area and let £, £/, £ be the represent
atives thereon of the places xlt a:/, x. Then the correct statement is that
is a continuous function of ar/ or |/ up to and including the branch place a^.
This is in fact the form in which the function n*1''*2 arises in the proof
X, C
of its existence upon which our account is based*.
In a similar way the function
-p*. c
regarded as a function of #/, is such that
is a finite continuous function of £' in the immediate neighbourhood of x.
* The reader may consult Neumann, p. 220.
B- 2
18 ONE INFINITY AT A BRANCH PLACE. [17
17. It may be desirable to give some simple examples of these integrals.
(a) For the surface represented by
y*=x(x-al)...(x-aap + l),
wherein alt ..., a2p + i are a^ finite and different from zero and each other, consider the
integral
i (dxfy+ri y+m\
^ J y \*-k *-&/
(£> "?)) (£i> »?i) being places of the surface other than the branch places, which are
(0, 0),(alt 0), ..., (a2p + 1, 0).
It is clearly infinite at these places respectively like log (x - £), - log (x - £ t).
It is not infinite at (£, -r,), (&, -7l); for (y + ?)/(# - £), (y + ih)/(* - &) are finite at
these places respectively.
At a place #=00 , where .« = r1, y = ft-f~l (l+P^t}}, t being ±1, and P1(t) a series of
positive integral powers of t vanishing for t = 0, we have
and the integral has the form
A being a constant. It is therefore finite.
At a place y = 0, for instance where
B being a constant, the integral has the form
C
C being a constant, and is finite.
Thus it is an elementary integral of the third kind with infinities at (£, »/), (£1}
It may be similarly shewn that the integral
, [dx fy y + r)i\
*j^U~^rJ
is infinite at (|1} j^) like — log(.r- £j) and is not elsewhere infinite except at (0, 0).
Near (0, 0), we have x=P,y = Dt [1 +P5 (t2)] and this integral is infinite like
Cdt
It is therefore an elementary integral of the third kind with one infinity at the
branch place (0, 0) and the other at (glt rjj).
Consider next the integral
(dx d
where rf = -^. It can easily be seen that it is not infinite save at (£, 17). Writing for the
ag
neighbourhood of this place, which is supposed not to be a branch place,
17] ONE INFINITY AT A DOUBLE POINT. 19
the integral becomes
(_dx
](x-
which is equal to
Thus the integral is there infinite like -- ^, and is thus an elementary integral of
x~ £
the second kind.
The elementary integral of the second kind for a branch place, say (0, 0), is a multiple of
»/*.
2 ]xy
In fact near # = 0, writing x=tz, y = Dt[l +P(t2)], this integral becomes
which is equal to
as desired.
The integral is clearly not infinite elsewhere.
Example 1. Verify that the integral last considered is the limit of
y~
y L#-f
as the place (£, rf) approaches' indefinitely near to (0, 0).
Example 2. Shew that the general integral of the first kind for the surface is
[dx I A A A _n
y 1 P-I •
(/9) We have in the first chapter §§ 2, 3 spoken of a circumstance that can arise, that
two sheets of the surface just touch at a point and have no further connexion, and we
have said that we regard the points of the sheets as distinct places. Accordingly we may
have an integral of the third kind which has its infinities at these two places, or an integral
of the third kind having one of its infinities at one of these places. For example, on the
surface
/(#» y) = (y- »h#) (y - m??) + (#, y)3 + (x, y\ = 0
where (x, y)3, (.?:, y\ are integral homogeneous polynomials of the degrees indicated by the
suffixes, with quite general coefficients, and ml, mz are finite constants, there are at #=0
two such places, at both of which y = 0.
In this case
dx
f'(yY
where f(y) = g- , is a constant multiple of an integral of the third kind with infinities at
these two places (0, 0) ; and
'-mlx + A x2 +Bxy + Cyz dx
2—2
20 EXAMPLES. [17
is a constant multiple of an integral of the third kind, provided A , B, C be so chosen that
y — iri]X-\- Ax2 + Bxy + Cy2 vanishes at one of the two places other than (0, 0) at which
Lx+My is zero. Its infinities are at (i) the uncompensated zero of Lx + My which is not
at (0, 0), (ii) the place (0, 0) at which the expression of y in terms of x is of the form
y = m^x + Px2 + Qx3 + ...
In fact, at a branch place of the surface where x = a + t'2, f'(y) is zero of the first order,
[ dx
and dx=2tdt; thus I-^TT-^ is finite at the branch places. At each of the places (0, 0),
f(y] is zero of the first order, Lx + My is zero of the first order and y - m^x -f- A x2 + Exy -f Oy2
is zero at these places to the first and second order respectively. These statements are
easy to verify ; they lead immediately to the proof that the integrals have the character
enunciated.
The condition given for the choice of A , B, C will not determine them uniquely — the
integral will be determined save for an additive term of the form
dx
'f'(yY
where P, Q are undetermined constants. The reader may prove that this is a general
integral of the first kind. The constants P, Q may be determined so that the integral of
the third kind has no periods at the period loops of the first kind, whose number in this
case is two. The reasons that suggest the general form written down will appear in the
explanation of the geometrical theory.
(•y) The reader may verify that for the respective cases
^/4 _ — ( /\t fy\ ( M J\\ //v» Ci
the general integrals of the first kind are
fdx , * w ^
I _ (3C 0) (3C C) *
Jy6
'dx .
—z(x~c^
I
—a(x- c)2 [A y2 + By (x - c) + C (x - c)2],
f
where A, B, C are arbitrary constants.
See an interesting dissertation "de Transformatione aequationis yn = R(x}.." Eugen.
Netto (Berlin, Gust. Schade, 1870).
(S) Ex. Prove that if F denote any function everywhere one valued on the Riemann
surface and expressible in the neighbourhood of every place in the form
the sum of the coefficients of the logarithmic terms log t of the integral / Fdx, for all
places where such a term occurs, is zero.
18] PERIODS OF INTEGRAL OF SECOND KIND. 21
It is supposed that the number of places where negative powers of t occur in the
expansion of F is finite, but it is not necessary that the number of negative powers be
finite. The theorem may he obtained by contour integration of I Fdx, and clearly
generalizes a property of the integral of the third kind.
18. The value of the integral* jr*'c dv*'° taken round the p closed curves
formed by the two sides of the pairs of period loops (alt b^\ ..., (av, bp\ in such
a direction that the interior of the surface is always on the left hand, is equal
to the value taken round the sole infinity, namely the place a, in a counter
clockwise direction. Round the pair ar, br the value obtained is
flr I dv*'C ,
taken once positively in the direction of the arrow head round what in the
figure is the outer side of br. This value is Qr(- a)ir), where a>ir denotes the
period of vt for transit of ar, namely, from what in the figure is the inside of
the oval ar to the outside.
The relations indicated by the figure for the signs adopted for wir, rir and
the periods of T*' ° will be preserved throughout the book.
Since a>ir is zero except when r = i, the sum of these p contour integrals
18 — <>>i ,i^i- Taken in a counter-clockwise direction, round the pole of F***
a '
where
the integral gives
- \ + A + Bt + CP + ...1 \Dv*c + t&va.'c + -...left,
where D denotes . Hence, as wit t = 1,
* Cf. Forsyth, pp. 448, 451. Harkness and Morley, p. 439.
22 ALL INTEGRALS AND RATIONAL FUNCTIONS [IS
This is true whether a be a branch place or a place at infinity (for which,
if not a branch place, x = t-1) or an ordinary finite place. In the latter case
. d ( x,
x,c\
v. .
* /
j-
dx\
Similarly the reader may prove that the periods of 11^' are
Orv
, ...... 0,
In this case it is necessary to enclose x± and xz in a curve winding Wi + 1
times at x1} w2 + 1 times at #2, in order that this curve may be closed.
19. From these results we can shew that the integral of the second kind
is derivable by differentiation from the integral of the third kind. Apart
from the simplicity thus obtained, the fact is interesting because, as will
appear, the analytical expression of an integral of the third kind is of the
same general form whether its infinities be branch places or not ; this is not
the case for integrals of the second kind.
We can in fact prove the equation
namely, if, to take the most general case, x± be a winding place and #/ a place
in its neighbourhood such that #/ = xl + t™ , the equation,
For, let the neighbourhood of the branch place xl be conformally represented
upon a simple closed area without branch place, by means of the infinitesimal
of x, as explained in the previous chapter. Let £/, & be the representatives
of the places #/, #1} and f the representative of a place x which is very near
to #!, but is so situate that we may regard #/ as ultimately infinitely closer
to #1 than x is.
Then x-x^ = (f - £)w+1,
where C does not vanish for #/ = x,
and E«!/*« = 1°£ (x ~ ^i') + 3*' = l°g (f - £0 + 9 >
where <£' is finite for the specified positions of the places and remains finite
when gi is taken infinitely near to £j (§ 16).
X C
Also II ' = n log (« — a,) + d> = log (£ — tj) 4- 9,
Xi. X* nil _i_lC>x ' ' O^3 ;>/ I
19] DERIVABLE FROM INTEGRAL OF THIRD KIND. 23
where <f> is also finite. Therefore
X,', a^ *„ x,
rn*',c - n*'c ~i i
im. -^r? — fc~^'~~ ' = ~~ e — fc
and thus
lim
where \/r is finite.
Now as £/ moves up to £ , for a fixed position of £, we have
i
fc ' _ fc — (T ' _ ™ yt!+l — /
?i <Ti — y^i *i/ ~ •*! >
and rx' e = r!1 ° = :L + «y,
%\ «1 ^ £j
where ^ is finite.
Hence Dtx H*' r - r*' c
is finite when x is near to a;^
Moreover it does not depend on #2. For from the equation
U*'c =I%'X2,
•*J\ j 2/2 •*'» "
we may regard H^' c^ as a function of xl , which is determinate save for an
additive constant by the specification of a; and c only. This additive constant,
which is determined by the condition that the function vanishes when x^ =xz,
is the only part of the function which depends on a?2. It disappears in the
differentiation.
Finally, by the determination of the periods previously given, it follows
that
has no periods at the 2p period loops. Hence it is a constant, and therefore
zero since it vanishes when x = c.
Corollary i.
Hence D,^ = I>tfDt^^D^Dtx^' = D,^'', \ :
of which neither depends on the constant position c.
Corollary ii.
The functions
24 PROOF FOR RATIONAL FUNCTIONS. [19
are respectively infinite like
111
tx 2 ' tx 3 ' tx 4 ' —
We shall generally write DXi, D2Xi, ... instead of Dtv , D\v ..... When XT
is an ordinary place DXi will therefore mean -=— , etc.
Corollary iii.
By means of the example (8) of § 17 it can now be shewn that the infinite
parts of the integral
\Fdx,
J
in which F is any uniform function of position on the undissected surface
having only infinities of finite order, are those of a sum of terms consisting of
proper constant multiples of integrals of the third kind and differential
coefficients of these in regard to the parametric place.
20. One particular case of Cor. iii. of the last Article should be stated.
A function which is everywhere one-valued on the undissected surface must
be somewhere infinite. As in the case of uniform functions on a single
infinite plane (which is the particular case of a Riemann surface for which
the deficiency is zero), such functions can be divided into rational and
transcendental, according as all their infinities are of finite order and of finite
number or not. Transcendental functions which are uniform on the surface
will be more particularly considered later. A rational uniform function can
be expressed rationally in terms of x and y*. But since the function can be
expressed in the neighbourhood of any of its poles in the form
A A A
n _L 1 _j_ 2 _i_ i •"•»».
T + ^+">+~r
we can, by subtracting from the function a series of terms of the form
obtain a function nowhere infinite on the surface and having no periods at the
first p period loops. Such a function is a constant f. Hence F can also be
expressed by means of normal integrals of the second kind only. Since F
has no periods at the period loops of the second kind there are for all rational
functions certain necessary relations among the coefficients Alt...,Am.
These are considered in the next Chapter.
* Forsyth, p. 369. Harkness and Morley, p. 262.
t Forsyth, p. 439.
21] SPECIAL RATIONAL FUNCTIONS. 25
21. Of all rational functions there are p whose importance justifies a
special mention here ; namely, the functions
dvi dv2 dvp
dx ' dx ' dx
In the first place, these cannot be all zero for any ordinary finite place a of
the surface. For they are, save for a factor 2?™', the periods of the normal
integral F*1 c. If the periods of this integral were zero, it would be a rational
uniform function of the first order; in that case the surface would be repre-
sentable conformally upon another surface of one sheet*, £= F/-6 being the
new independent variable ; and the transformation would be reversible
(Chap. I. § 6). Hence the original surface would be of deficiency zero ;
in which case the only integral of the first kind is a constant. The functions
are all infinite at a branch place a. But it can be shewn as here that the
quantities to which they are there proportional, namely J)avly ..., Davp, cannot
be all zero. The functions are all zero at infinity, but similarly it can be
shewn that the quantities, Dv1} ... , Dvp> cannot be all zero there.
Thus p linearly independent linear aggregates of these quantities cannot all vanish at
the same place. We remark, in connexion with this property, that surfaces exist of all
deficiencies such that p - 1 linearly independent linear aggregates of these quantities
vanish in an infinite number of sets of two places. Such surfaces are however special, and
their equation can be putf into the form
y = w "• /2P + 2 •
We have seen that the statement of the property requires modification
at the branch places, and at infinity ; this particularity is however due to the
behaviour of the independent variable x. We shall therefore state the pro
perty by saying: there is no place at which all the differentials dvlt ..., dvp
vanish. A similar phraseology will be adopted in similar cases. For instance,
we shall say that each of dvl} dv^, ... , dvp has| 2p — 2 zeros, some of which
may occur at infinity.
In the next place, since any general integral of the first kind
must necessarily be finite all over any other surface upon which the original
surface is conformally and reversibly represented and therefore must be an
integral of the first kind thereon, it follows that the rational function
dx p dx
* I owe this argument to Prof. Klein. + See below, Chap. V.
J See Forsyth, p. 461. Harkness ami Morley, p. 450.
26 INVARIANCE OF THEIR RATIOS. [21
is necessarily transformed with the surface into
dV
where Vi = Vt is an integral of the first kind, not necessarily normal, on the
new surface, f being the new independent variable, and M = ~ .
(LOG
Thus, the ratios of the integrands of the first kind are transformed
into ratios of integrands of the first kind ; they may be said to be invariant
for birational transformation.
This point may be made clearer by an example. The general integral
of the first kind for the surface
y- = (as, 1)8
can be shewn to be
'dx ,
y
A, B, C being arbitrary constants.
If then 0! : 0o : 03 denote the ratios of any three linearly independent
integrands of the first kind for this surface, we have
for proper values of the constants altbi, ... , c3,
and hence
Such a relation will therefore hold for all the surfaces into which the given
one can be birationally transformed.
22. It must be remarked that the determination of the normal integrals
here described depends upon the way in which the fundamental period loops
are drawn. An integral of the first kind which is normal for one set of
period loops will be a linear function of the integrals of the first kind which
are normal for another set ; and an integral of the second or third kind, which
is normal for one set of period loops, will for another set differ from a normal
integral by an additive linear function of integrals of the first kind.
27
CHAPTER III.
THE INFINITIES OF RATIONAL UNIFORM FUNCTIONS.
23. IN this chapter and in general we shall use the term rational function
to denote a uniform function of position on the surface of which all the
infinities are of finite order, their number being finite. We deal first of all
Avith the case in which these infinities are all of the first order.
If k places of the surface, say a^, a2 •••«*, be arbitrarily assigned we can
always specify a function with p periods having these places as poles, of the
first order, and otherwise continuous and uniform ; namely, the function is of
the form
where the coefficients /*0, /^ ... /A^ are constants, the zeros of the functions F
being left undetermined. Conversely, as remarked in the previous chapter
(§ 20), a rational function having a,, ..., a^ as its poles must be of this form.
In order that the expression may represent a rational function the periods
must all be zero. Writing the periods of F£ in the form fij (a), ...,£lp (a),
this requires the equations
(cr.a) + . . . + A**n,- (at) = 0,
for all the p values, i = 1, 2, . . . , p, of i. In what follows we shall for the sake
of brevity say that a place c depends upon r places c1} c2, ..., cr when for all
values of i, the equations
fli (C) =f&i (C,) + . . . +/A (C,.)
hold for finite values of the coefficients fi,--',fr, these coefficients being
independent of i. Hence we may also say :
In order that a rational function should exist having k assigned places as
its poles, each simple, one at least of these places must depend upon the others.
24. Taking the k places c^, a2, ..., a* in the order of their suffixes, it may
of course happen that several of them depend upon the others, say a,+i, ...,««,-
28 DEPENDENCE OF POLES OF A RATIONAL FUNCTION [24
upon ttx, ..., as, the latter set an . .., as being independent: then we have
equations of the form
,+lf
fti (a*) = nt, ! fti (aO + . . . + nk> , Of (a,),
the coefficients in any of the rows here being the same for all the p values of
i. In particular, if s be as great as p and alt ... , as be independent, equations
of this form will hold for all positions of as+1, ..., ak. For then we have
enough disposeable coefficients to satisfy the necessary p equations.
When it does so happen, that a8+1,...,ak depend upon 0,1... at, there
exist rational functions, of the form
i —
wherein cr4+1 ... o-^, Xs+1 ... X^ are constants, which are all infinite once in
ttj ... as and are, beside, infinite respectively at as+1) ..., a^ ; and the most
general function uniform on the dissected surface, which is infinite to the
first order at a1, . . . , a^ , being, as remarked, of the form
/*<> + PI r 4-
can be written in the form
4
+ /%|r- ^* -hn*. i It, + ...... +nktS F^-
\_^k
namely, in the form
v0 + z/ir*i + ...... + v, F^ + vg+1Rt+1 + ...... +vkRk.
If this function is to have no periods, the equations
vini(al) + ...... + v.n» (a.) = 0, (i=l, 2, ...,p),
must hold. Since a1} ...,as are independent, such equations can only hold
when Vi = 0 = . . . = vg. Thus the most general rational function having k
poles of the first order, at a1} ..., a*, is of the form
i/o + vs+lJRg+1 + ...... + vkRk,
and involves k —s+ I linearly entering constants, s being the number of
places among alf ... , ak which are independent. These constants will generally
be called arbitrary : they are so only under the convention that a function
25] DETERMINES EXPRESSION OF FUNCTION. 29
which has all its poles among a1} ...,ak be reckoned a particular case of a
function having each of these as poles ; for it is clear that, for instance, Rk is
only infinite at a1( ..., at, ak. The proposition with a slightly altered enuncia
tion, given below in § 27 and more particularly dealt with in § 37, is called
the Riemann-Roch Theorem, having been first enunciated by Riemann*,
and afterwards particularized by Rochf.
25. Take now other places ak+l, ak+2, ... upon the surface in a definite
order, and consider the possibility of forming a rational function, which beside
simple infinities at alt ..., ak has other simple poles at, say, ak+1, ak+z, ...,ah.
By the first Article of the present chapter it follows that the least value
of h for which this will be possible will be that for which ah depends
on ch ... akak+l ... a/,-i, that is, depends on a^ . . . as ak+l . . . «&_!. This will
certainly arise at latest when the number of these places a^ ... as ak+l . . . ah-i
is as great as p, namely h — l=k + p — s, and if none of the places ak+l . . . «/,_!
depend upon the preceding places ax ... as, it will not arise before: in that
case there will be no rational function having for poles the places
ak+j
for any value of j from 1 to p — s.
But in order to state the general case, suppose there is a value of j less
than or equal to p — s, such that each of the places
ak+j+i ...... ah
depends upon the places
the smallest value of j for which this occurs being taken, so that no one of
ak+1 . . . ak+j depends on the places which precede it in the series
Then there exists no rational function with its poles at a, ... ak ak+l ... ak+j,
but there exist functions
ia
,s 1 as ~ nk+j+i,k+i 1 ak + l ~ ...... ~ nk+}+i,k+j 1 ak + i J >
whose poles are respectively at
for all values of i from 1 to h — k — j.
* Riemann, Ges. Werke, 1876, p. 101 (§ 5) and p. 118 (§ 14) and p. 120 (§ 16).
t Crelle, 64. Cf. also Forsyth, pp. 459, 464. The geometrical significance of the theorem
has been much extended by Brill and Noether. (Math. Ann. vii.)
30 STATEMENT OF COMPLETE RESULT. [25
Then the most general rational function with poles at
is in fact
and involves k — s + i + 1 arbitrary constants, namely the same number as
that of the places of the set
which depend upon the places that precede them.
For such a function must have the form
-1- P-k+j lak
namely,
— s
"i" ^ Ps+r ^ -tts+r T ^s+r, I *• «, T ...... T ^s+r,s *• a, C
r=l L^-«+r ^-s+r
1 T&
AfcffH + nk+}+t,i !«, + ......
H
r"31 . rix n*
t, s 1 a, + Vk+j+t, k+i i ak + 1 + ...... + Kk+j+t, k+j A ak +,
which is of the form
V0 + Vi t\^ + ...... + V8
+ vk+l r^ + 1 + ......
and the p periods of this, each of the form
Vi H (ttj) + ...... + i/»ft (a,.) + i>t+1 ft
cannot be zero unless each of vl ... vsvk+i ••• »k+j be zero, for it is part of
the hypothesis that none of ak+1 . . . ak+j depend upon preceding places.
26. Proceeding in this way we shall clearly be able to state the following
result —
Let there be taken upon the surface, in a definite order, an unlimited
number of places al} a2, — Suppose that each of al...a,Q_ is inde
pendent of those preceding it, but each of a^,^ ... aQi depends on
a, ... a« Suppose that each of an , , a~ ^ . . an is independent of
Ifc—fl Vi + l Vi + 4 Va~9i
those that precede it in the series a, ... an an,....an but each of
VI -9i 9fT* V2~9a'
aQ,-^i ••• % dePends uP°n «i ••• aQl-qiaQl+i ••• aQt.qt' This requires that
26] EXPRESSION OF FUNCTION OF ASSIGNED POLES. 31
Suppose that each of aQ +l . . . aQ _ is independent of those that precede it
in the series a, ... a~ a~ , . . . . an «„ , . . . . an , but each of an ..... an
Qi-9i Ci+l Qi-<lt Qs+l Qi-Qs §3-93+1 Qa
depends upon the places of this series. This requires that
Qi-qi + [Q2-q^-Qi] + [Q3-q3-Q2]>p-
Let this enumeration be continued. We shall eventually come to places
aQ +i'aQ +2' "• ao - ' eac^ ^dependent of the places preceding, for which
the total number of independent places included, that is, of places which
do not depend upon those of our series which precede them, is p — so that
the equation
will hold. Then every additional place of our series, those, namely, chosen
in order from aQ _ +l,aQ _ +2, ... will depend on the preceding places of the
whole series.
This being the case, it follows, using Rf as a notation for a rational
function having its poles among al ... a/, that rational functions
do not exist.
The number of these non-existent functions is p.
For all other values off, a rational function Rj exists.
To exhibit the general form of these existing rational functions in the
present notation, let m be one of the numbers 1, 2, ..., h; i be one of the
numbers 1, 2, ... qtn, and let the dependence of aQ _ . upon the preceding
places arise by p equations of the form
then, denoting P* by F,., there is a rational function
which has its poles at
a' •'• aft-7,' %+!••• aQ.>-<,.>> •— aQn
and the general rational function having its poles at
~'a<jm-9m>
32 THERE ARE p GAPS. [26
is of the form
and involves ql + qz + . . . 4- <?m_i + i + 1 arbitrary coefficients.
The result may be summarised by putting down the line of symbols
(&-?, + !),..,&, & + !,... ,&-, + !, ...,(&-?*), (&-?*+!),...
with a bar drawn above the indices corresponding to the places which depend
upon those preceding them in the series. The bar beginning over Qh — qh + I
is then continuous to any length. The total number of indices over which
no bar is drawn is p. There exists a rational function Rf, in the notation
above, for every index which is beneath a bar.
The proposition here obtained is of a very fundamental character. Sup
pose that for our initial algebraic equation or our initial surface, we were able
only to shew, algebraically or otherwise, that for an arbitrary place a there
exists a function Kxa, discontinuous at a only and there infinite to the first
order, this function being one valued save for additive multiples of & periods,
and these periods finite and uniquely dependent upon a, then, taking arbitrary
places a1} a2, ... upon the surface, in a definite order, and considering func
tions of the form
that is, functions having simple poles at al} ..., a#, we could prove, just as
above, that there are k values of N for which such functions cannot be one
valued ; and obtain the number of arbitrary coefficients in uniform functions
of given poles. Namely, the proposition would furnish a definition of the
characteristic number k — which is the deficiency, here denoted byp — based
upon the properties of the uniform rational functions.
We shall sometimes refer to the proposition as Weierstrasss gap
theorem*.
27. When a place a is, in the sense here described, dependent upon places
bi} 62, ... , br, it is clear that of the equations
* " Liickensatz." The proposition has been used by Weierstrass, I believe primarily under
the form considered below, in which the places ax, a2, ... are consecutive at one place of the
surface, as the definition of p. Weierstrass's theory of algebraic functions, preliminary to a theory
of Abelian functions, is not considered in the present volume. His lectures are in course of
publication. The theorem here referred to is published by Schottky : Conforme Abbildung
mehrfach zusammenhangender ebener Flachen, Crelle Bd. 83. A proof, with full reference to
Schottky, is given by Noether, Crelle Bd. 97, p. 224.
27] TRANSPOSITION OF THE LINEAR CONDITIONS. 33
A.n, (br) + . . . + Apfip (br) = o
A.fl^a) +... + Apflp(a) = 0
the last is a consequence of those preceding — and conversely that when the
last equation is a consequence of the preceding equations the place a depends
upon the places b1} b2, ..., br.
Hence the conditions that the linear aggregate
0 (as) = A& (as) + . . . + Apnp (as)
should vanish at the places
^"•aQlaQi+l--'aQ3aQ,+l'"aQm-<)a+i'
wherein i$> qm, are equivalent to only
or
linearly independent equations.
If then r + 1 be the number of linearly independent linear aggregates of
the form £1 (as), which vanish in the Qm - qm -f i specified places, we have
T + 1 =p - (Qm - ql - ... - qm).
Denoting Qm — qm + i by Q, and the number of constants in the general
rational function with poles at the Q specified places, of which constants one
is merely additive, by q + 1,
q + 1 = q, + q2 + ... + qm^ + i + 1.
We therefore have
Q-q=p-(r + i).
Recalling the values of fl^ar)... Clp(x) and the fact (Chapter II. § 21)
that every linear aggregate of them vanishes in just 2p - 2 places, we see
that when Q is greater than 2p - 2, T + 1 is necessarily zero.
In the case under consideration in the preceding article the number
T + 1 for the function EQ , namely the number of linearly independent
linear aggregates ft (as) which vanish in the places
is given, by taking m = h-l and i = qh_, in the formula of the present
article, by the equation
r + 1 = p -
= Qh
B.
34 POLES AT ONE PLACE. [27
Hence one such linear aggregate vanishes in the places
and therefore
&-?*-! >2p-2
or, the index associated with the last place aQ _ of our series, corresponding to
^h 'k
which a rational function RQ _ does not exist, is not greater than 2p — 1. A
^A ^A
case in which this limit is reached, which also furnishes an example of the
theory, is given below § 37, Ex. 2.
28. A limiting case of the problem just discussed is that in which the
series of points a1} a.2, ... are all consecutive at one place of the surface.
A rational function which becomes infinite only at a place, a, of the
surface, and there like
GI GZ Cf
t P tr '
where any of the constants Glt C2, ... Cr_lt but not Cr, maybe zero, t being the
infinitesimal, is said to be there infinite to the rth order. If— A.t- = G[/(i — 1)!,
such a function can be expressed in a form
x + XjF* + x2z>ar* + ... + vo^rs
where, in order that the function be one valued on the undissected surface,
the p equations
X, flf (a) + \2Da nf (a) + . . . + X^"1 «; (a) = 0
must be satisfied : and conversely these equations give sufficient conditions
for the coefficients X1; Xg, ... , X,..
In other words, since Xr cannot be zero because the function is infinite to
the ?'th order, the p differential coefficients D^~lCli(a), each of the r— 1th
order, must be expressible linearly in terms of those of lower order,
with coefficients which are independent of i. We imagine the p quantities
Du~1fli(a), for i = l, 2, ...,p, written in a column, which we call the rth
column ; and for the moment we say that the necessary and sufficient con
dition for the existence of a rational function, infinite of the rth order at a,
and not elsewhere infinite, is that the rth column be a linear function
of the preceding columns.
Then as before, considering the columns in succession, they will divide
themselves into two categories, those which are linear functions of the pre
ceding ones and those which are not so expressible. And, since the number
of elements in a column is p, the number of these latter independent columns
30] CORRESPONDING TRANSPOSITION OF LINEAR CONDITIONS. 35
will be just p. Let them be in succession the ^th, &2th, ...,kpt\\. Then
there exists no rational function infinite only at a, and there to these
orders klt k2, ..., kp, though there are integrals of the second kind infinite
to these orders. But if Q be a number different from klt ..., kp, there does
exist such a rational function of the Qth order, its most general expression
being of the form
xQD(?-ir* + XQ-xA?-2^ + ... + xxr* + x,
namely, the integral of the second kind whose infinity is of order Q is
expressible linearly by integrals of the second kind of lower order of infinity,
with the addition of a rational function.
If q + 1 be the number of linearly independent coefficients in this function,
one being additive, we have an equation
Q-q=P-(r + i),
where p — (r + 1) is the number of the linearly independent equations of the
form
\iflf (a) + X2Z)nt-(a) + ... + \QD^fli (a) = 0, (i = 1, 2, ..., p),
from which the others may be linearly derived. As before, r + I is the
number of linearly independent linear aggregates of the form
which satisfy the Q conditions
A.D^, (a) + ... + ApDrnp (a) = 0
forr = 0, 1,2, ...,Q-1.
29. In regard to the numbers ^ . . . kp we remark firstly that, unless p = 0,
&! = 1 — for if there existed a rational function with only one infinity of the
first order, the positive integral powers of this function would furnish rational
functions of all other orders with their infinity at this one place, and there
would be no gaps (compare the argument Chapter II. § 21); and further
that in general they are the numbers 1, 2, 3 ... p, that is to say, there is only
a finite number of places on the surface for which a rational function can be
formed infinite there to an order less than p + 1 and not otherwise infinite.
We shall prove this immediately by finding an upper and a lower limit to
the number of such places (§ 31).
30. Some detailed algebraic consequences of this theory will be given in
Chapter V. It may be* here remarked, what will be proved in Chapter VI.
in considering the geometrical theory, that the zeros of the linear aggregate
It is possible that the reader may find it more convenient to postpone the complete
discussion of § 30 until after reading Chapter vi.
3—2
36 ILLUSTRATION FROM THE SUBSEQUENT [30
can be interpreted in general as the intersections of a certain curve, of the
form
</> = A^ (x) + ...+ Ap(f>p O) = 0,
wherein ^...^>p are integral polynomials in x and y, with the curve repre
sented by the fundamental equation of our Riemann surface. In such
interpretation, the condition for the existence of a rational function of order Q
with poles only at the place a, is that the fundamental curve be of such
character at this place that every curve <£, obtained by giving different values
to A1... Ap, which there cuts it in Q— 1 consecutive points, necessarily cuts
it in Q consecutive points. As an instance of such property, which seems
likely also to make the general theory clearer, we may consider a Riemann
surface associated with an equation of the form
f(x, y) = K + (x, y\ + (x, y\ + (x, y)3 + (x, y\ = 0,
wherein (x, y)r is a homogeneous integral polynomial of the rth degree, with
quite general coefficients, and K is a constant. Interpreted as a curve, this
equation represents a general curve of the fourth degree ; it will appear
subsequently that the general integral of the first kind is
dx (A+Bx+Cy),
where f (y) = df/dy, and A , B, 0 are arbitrary constants ; and thence, if we
recall the fact that flj (as), ...,£lp(x) are differential coefficients of integrals
of the first kind, that the zeros of the aggregate
may be interpreted as the intersections of the quartic with a variable straight
line.
Take now a point of inflexion of the quartic as the place a. Not every
straight line there intersecting the curve in one point will intersect it in any
other consecutive point ; but every straight line there intersecting the curve
in two consecutive points will necessarily intersect it there in three consecu
tive points. Hence it is possible to form a rational function of the third
order whose only infinities are at the place of inflexion ; in fact, if
be the equation of the inflexional tangent, and
be the equation of any line through the fourth point of intersection of the
inflexional tangent with the curve, the ratio of the expressions on the left
hand side of these equations, namely
Ax + By + 1
30] GEOMETRICAL THEORY. 37
is a general rational function of the desired kind, as is immediately obvious
on consideration of the places where it can possibly be infinite. Thus for the
inflexional place the orders of two non-existent rational functions are 1, 2.
It can be proved that in general there is no function of the fourth order — the
gaps at the orders 1, 2, 4 are those indicated by Weierstrass' theorem.
In verification of a result previously enunciated we notice that since
Ax + By+ 1 = 0 may be taken to be any definite line through the fourth
intersection of the inflexional tangent with the curve, the function contains
# + 1 = 2 arbitrary constants. From the form of the integrals of the first
kind which we have quoted, it follows that p = 3 ; thus the formula
wherein Q = 3, requires r + 1 = 1 ; now by § 28 r + 1 should be the number
of straight lines which can be drawn to have contact of the second order with
the curve at the point : this is the case.
If the quartic possess also a point of osculation, a straight line passing
through two consecutive points of the curve there will necessarily pass
through three consecutive points and also necessarily through four. Hence,
for such a place, we can form a rational function of the third order and one
of the fourth. In fact, if A<p + B0y + 1=0 be the tangent at the point of
osculation and A^x + BJJ + 1 = 0 be any other line through this point, while
\£c + fj,y+v = 0 is any other line whatever, these functions are respectively,
in their most general forms,
A^x + B$ + 1 Xx + fjuy +j/
' + ** A«x + B0y + 1 ' A^i+B0y + l '
wherein X, p, v are arbitrary constants.
It can be shewn that in general we cannot form a rational function of the
fifth order whose only infinity is at the place of osculation. Thus the gaps
indicated by Weierstrass's theorem occur at the orders 1, 2, 5. (Cf. the
concluding remark of § 34.)
In case, however, the place a be an ordinary point of the quartic, the
lowest order of function, whose only infinity is there, is p + 1 = 4 : it will
subsequently become clear that a general form of such a function in S'/S,
where S = 0 is any conic drawn to intersect the quartic in four con
secutive points at a, and S' = 0 is the most general conic drawn through
the other four intersections of S with the quartic. S' will in fact be of the
form \S + p,T, where T is any definite conic satisfying the conditions for S',
and X, /j, are arbitrary constants; the equation Q— q=p — (r + 1) is clearly
satisfied by Q = 4, q = 1, p = 3, T + 1 = 0.
The present article is intended only by way of illustration ; the examples
given appear to find their proper place here. The reader will possibly
38
FUNCTION OF ORDEK
[30
find it desirable to read them in connexion with the geometrical account
given in Chapter VI.
31. Consider now what places of the surfaces are such that we can form
a rational function infinite, only there, to an order as low as p.
For such a place, as follows from § 28, the determinant
A =
> (r\ DP"1 O (v\ T}P— ! O ff\
4i v*v> ^ i£2vv> >-L/ ***v~y
must vanish. Assume for the present that none of the minors of A vanish
at that place. It is clear by § 28 that A only vanishes at such places as we
are considering.
Let v be any integral of the first kind. We can write
/ \ dvi . , „ dv dvi
(x) = -j- in the form -=- — ,
at at av
and similarly put
and so write
where D is the determinant whose rth row is formed with the quantities
dvr '
' dvr
Now -T* is a rational function; and it is infinite only at the zeros of dv,
whose aggregate number is 2p — 2; and -y-0* is a rational function of the
(4>p — 4)th order, its poles being also at the zeros of dv; and a similar state
ment can be made in regard to the other rows of D.
Hence D is a rational function whose infinities are of aggregate number
(2p - 2) (1 + 2 + ... +p) = (p - l)p (p + 1),
and this is therefore the number of zeros of D.
32] EXISTS ONLY FOR CERTAIN PLACES. 39
Now A can vanish either by the vanishing of the factor D or by the
(dv\%p (.P+I)
— I The zeros of the last factor are, however,
dtj
the poles of D. Hence the aggregate number of zeros of A is (p — 1) p (p + 1).
We shall see immediately that these zeros do not necessarily occur at as
many as (p — I)p (p + 1) distinct places of the surface.
In order that a rational function should exist of order less than p, its
infinity being entirely at one place, say of order p — r, it would be necessary
that the r determinants formed from the matrix obtained by omitting the
last r rows of A should all vanish at that place. We can, as in the case of
A, shew that each of these minors will vanish only at a finite number of
places. It is therefore to be expected that in general these minors will not
have common zeros ; that is, that the surface will need to be one whose
3/; — 3 moduli are connected in some special way.
Moreover it is not in general true that a rational function of order p + 1
exists for a place for which a function of order p exists, these functions not
being elsewhere infinite. For then we could simultaneously satisfy the two
sets of p equations
(a) + \DCli (a) + ...... + Xp^DP-2^ (a) + \pDf~1 Ot- (a) = 0,
(a) + ^Dtli (a) + ...... + ffv-lDP-afli (a) + /v,., jDPflf (a) = 0,
namely, A and -7- would both be zero at such a place. The condition that
at
this be so would require that a certain function of the moduli of the
surface — what we may call an absolute invariant — should be zero.
Therefore when of the p gaps required by Weierstrass's theorem, p — 1
occur for the orders 1, 2, ..., p — 1, the other will in general occur for the
order p+l. The reader will see that there is no such reason why, when a
function of order p exists, a function of order p + 2 or higher order should
not exist.
32. The reader who has followed the example of § 30 will recall that the
number of inflexions of a non-singular plane quartic* is 24 which is equal to
the value of (p - 1) p (p + 1) when p = 3. The condition that the quartic
possess a point of osculation is that a certain invariant should vanish^.
When the curve has a double point, there are only two integrals of the
first kind J, and p is equal to two. Thus in accordance with the theory above,
there should be (p — 1) p (p + 1) = 6 places for which we can form functions
* Salmon, Higher Plane Curves (1879), p. 213.
t The equation can be written so as to involve only 5 = 3/> - 3 - 1 parametric constants
(Chap. V. p. 98, Exs. 1, 2).
+ Their forms are given Chapter II. § 17 /3. lleasons are given in Chapter VI. The reader
may compare Forsyth, p. 395.
40 EXAMPLES. [32
of the second order infinite only at one of these places. In fact six tangents
can be drawn to the curve from the double point : if A^c + B9y = 0 be the
equation of one of these and \ (Ax + By) + fi(A0x + B0y) = 0 be the equation
of any line through the double point, the ratio
fc Ax + By
XA^ + B0y + f*
represents a function of second order infinite only at the point of contact of
For the point of contact of one of these tangents the^) gaps occur for the
orders 1 and 3.
The quartic with a double point can be biratioually related to a surface expressed by
an equation of the form
£ being the function above. The reader should compare the theory in Chapter I. and the
section on the hyperelliptic case, Chapter V. below.
33. Ex. For the surface represented by the equation
where the brackets indicate general integral polynomials of the order of the suffixes, p is
equal to 4, and the general integral of the first kind is
r
where f(y ) = + . Prove that at the (p - 1 ) p (p + 1 ) = 60 places for which rational functions
of the 4th order exist, infinite only at these places, the following equations are satisfied
2/7</-3(/'/y)2=o,
2f 3 s3/ ff2 83/,3i
>* + 6 a^ap fvfx ~ ap J* J
Where y'=' etc-'^=' etc>
Explain how to express these functions of the fourth order.
Enumerate all the zeros of the second differential expression here given.
Ex. 2. In general, the corresponding places are obtained by forming the differential
equation of the pth order of all adjoint <f> curves. In a certain sense A is a differential
invariant, for all reversible rational transformations. (See Chapter VI.)
* Here the number of integrands of the integrals of the first kind, which are of the form
(Lx + My)lf'(y) (cf. Chapter III. § 28), which vanish in two consecutive points at the point of
contact of Avx + H0y = 0, is clearly 1, or T + 1 = 1 : hence the formula Q - q -p - (r + 1) is verified
by Q = 2, q = l, p = 2, so that the form of function of the second order given in the text is the
most general possible.
34]
CONSIDERATION OF THESE EXCEPTIONAL PLACES,
41
34. We pass now to consider whether the (p — 1) p (p + 1) zeros of A
will in general fall at separate places*.
Consider the determinant
V = 0 fl (x} fl (x)
wherein flj^ ((?)=* J5f Of (£•), and k1} ..., kp are the orders of non-existent
rational functions for a place £, in ascending order of magnitude, (A^ = 1) ;
and let its value be denoted by
so
that ur = I (i>r (x) dtx is an integral of the first kind.
Then wr(x) vanishes at % to the (kr — l)th order.
For <w,. (x) is the determinant
v.-r-i
now the (kr — l)th differential coefficient of this determinant (in regard to
the infinitesimal at x) has at £ a value which is in fact the minor of the
element (1, 1) of V, save for sign. That this minor does not vanish is part
of the definition of the numbers k1} k2, ..., kp. But all differential coeffi
cients of Vr of lower than the (kr — l)th order do vanish at £: some, because
for a; = £ they are determinants having the first row identical with one of
the following rows, this being the case for the differential coefficients of
orders ^ — 1, &2 — 1, ... ; others, because when /* is not one of the numbers
fcj, k2, ..., kp, D^ifli^) is a linear function of those of Dk'~1Cli(^),
Lb~* {li(g), ... for which p is greater than klf k.2, ... , the coefficients of the
linear functions being independent of i. This proves the proposition.
It is clear that the &rth differential coefficient of Vr may also vanish at £.
In particular Wi(x) does not vanish at £: a result in accordance with a
remark previously made (Chapter II. § 21), that there is no place at which
the differentials of all the integrals of the first kind can vanish.
* The results in §§ 34, 35, 36 are given by Hunvitz, Math. Annul. 41, p. 409. They will
be useful subsequently.
42 AND OF THE NON-EXISTING-ORDERS. [34
An important corollary is that the highest order for which no rational
function exists, infinite only at the place £, is less than 2p. For wp (x) vanishes
only 2p — 2 times, namely, kp — 1 < 2p — 2.
35. We can now prove that if k2 > 2, the sum of the orders k\, &2> ••• , kp
is less than p2. For if there be a rational function of order in, infinite only
at £, and r be one of the non-existent orders* ^ ... kp, r — m is also one of
these non-existent orders — otherwise the product of the existent rational
function of order i — m with the function of order m would be an existent
function of order r. The powers of the function of order m are existent
functions, hence none of kl . . . kp are divisible by in.
Let Ti be the greatest of the non-existent orders k^ ... kp which is con
gruent to i(< m) for the modulus m : then, by the remark just made,
TI, Ti — in, Ti — 2m, ... , m + i, i
are all non-existent orders — and all congruent to i for the modulus m. Since
i'i occurs among ki...kp, all these also occur. Take i in turn equal to
1, 2, ... 771-1.
Then, the number of non-existent orders being p,
p =
so that T! + r2 + . . . + rm_^ = mp — \ m (m — 1)
= \ m(2p — m+ 1).
Now the sum of the non-existent orders is
m-l
2 [ri + (n - m) + (n - 2m) + ... + i] ,
which is equal to
+ lm (m - 1) - TL (in - 1) (2m - 1),
and, since Sr^ = ^ m (%p — m + 1), this is equal to
~ Sr, [r, - (2p - 1)] + I [4p» - (m - Vf\ + ^ (m - 1) (m + 1),
or ^-r2-l-r-m-lm-2.
* i.e. orders of rational functions, infinite only at £, which do not exist: and similarly in
what follows.
36]
LEAST NUMBER OF THESE EXCEPTIONAL PLACES.
Since, by the corollary of the preceding article, 2p — 1 is not less than riy
this is less than p- unless m is 1 or 2. Now m cannot be equal to 1 ; and if
it is 2 then also k2 > 2. Hence the statement made at the beginning of the
present Article is justified.
When there is a rational function of order 2, it is easy to prove that
there are places for which L\ ... kp are the numbers 1, 3, 5, ... , 2p — 1, whose
sum* is^>2. An example is furnished by § 32 above.
Ex. For the surface
for which p = 3, there is, at #=ao , only one place, and the non-existent orders are 1, 2, 5 :
whose sum is p* — l.
36. We have in § 34 defined p integrals of the first kind
I wl(x)dtx, ... , I wp(x)dtx
by means of a place £. Since the differential coefficients of these vanish at £
to essentially different orders, these integrals cannot be connected by a homo
geneous linear equation with constant coefficients. Hence a linear function
of them with parametric constant coefficients is a general integral of the first
kind. Therefore each of ^(x) ... O^ (x) is expressible linearly in terms of
o>! (x) ... wp (x) in a form
&i 0) = Cn$r(» + . . . + Cip&p (x},
where the coefficients are independent of x. Thus the determinant A (§ 31),
which vanishes at places for which functions of order less than p + 1 exist, is
equal to
>i(x) , ...... , <op(x)
xwl(x) , ...... , DXG>P(X)
where C is the determinant of the coefficients c,-j. It follows from the result
of § 34 that the determinant here multiplied by C vanishes at £ to the order
Thus, the determinant A vanishes at any one of its zeros to an order equal
to the sum of the non-existent orders for the place diminished by %p(p + l).
For example, it vanishes at a place where the non-existent orders are
1, 2, ... , p- 1, p + 1 to an order $p(p-~L)+p + l-^p(p + l) or to the
first order. We have already remarked that such places are those which
most usually occur.
* Cf. Burkhardt, Math. Annal. 32, p. 388, and the section iu Chapter V., below, on the hyper-
elliptic case.
44 RIEMANN-ROCH THEOREM. [36
Hence, since fa -f . . . + kp ^ p2, A vanishes at one of its zeros to an order
Further, if r be the number of distinct places where A vanishes, and
mly m2, . .., mr be the orders of multiplicity of zero at these places, it follows,
from
and raa + ... + mr < r %p(p — 1),
that r > 2p + 2, or
there are at least 2p + 2 distinct places for which functions of less order
than p + 1, infinite only thereat, exist] this lower limit to the number of
distinct places is only reached when there are places for which functions of
the second order exist.
Ex. For the surface given by
p is equal to 3 ; there are 12 = 2^ + 6 distinct places where A vanishes.
37. We have called attention to the number of arbitrary constants con
tained in the most general rational function having simple poles in distinct
places (§ 27) and to the number in the most general function infinite at a
single place to prescribed order (§ 28) : in this enumeration some of the con
stants may be multipliers of functions not actually becoming infinite in the
most general way allowed them, that is, either of functions which are not
really infinite at all the distinct places or of functions whose order of infinity
is not so high as the prescribed order.
It will be convenient to state here the general result, the deduction of
which follows immediately from the expression of the function in terms of
integrals of the second kind : —
Let tt1; a.,, ... be any finite number of places on the surface, the infinitesi
mals at these places being denoted by tl} t.2, .... The most general rational
function whose expansion at the place di involves the terms
JL JL _L
&' W t*<' '"
— whose number is finite, = Q» say, — and no other negative powers, involves
q + 1 linearly entering arbitrary constants, of which one is additive, q being
given by the formula
Q-q=P-(r + i),
where Q is the sum of the numbers Q{, and r + 1 is the number of linearly
independent linear aggregates of the form
fl(a;) = A A (a;) + ... + ApQ,p(x\
37] EXAMPLE. 45
which satisfy the sets of Qi relations, whose total number is Q, given by
A, DV-1 f^ (at) + A,D^ 02 (ai) + . . . + ApD*--1 np (tti) = 0,
Ail>-1 nt (a;) + 4a J> -i n2 (a<) + . . . + ^I> -1 Op (a,-) = 0,
As before, this general function will as a rule be an aggregate of functions
of which not every one is as fully infinite as is allowed, and it is
clear from the present chapter that in the absence of further information in
regard to the places a1} a.2, ... it may quite well happen that not one of these
functions is as fully infinite as desired, the conditions analogous to those stated
in §§ 23, 28 not being satisfied. See Example 2 below.
The equation Q — q=p — (r + l) will be referred to as the Riemann-Roch
Theorem.
Ex. 1. For a rational function having only simple poles or, more gene
rally, such that the numbers X;, /^t-, vi, ... for any pole are the numbers
1, 2, 3, ... Qit
if Q > 2p — 2, r + 1 is zero, since fl (x) has only an aggregate number
2p — 2 of zeros : the function involves Q — p + 1 constants,
if Q = 2p — 2, r + 1 cannot be greater than 1 ; for the ratio of two of the
aggregates £l(x) then vanishing at the poles, being expressible in a form
dV
_™ , where V, W are integrals of the first kind, would be a rational function
a w
without poles, namely a constant ; then the linear aggregates fl (#) would be
identical : thus the function involves Q — p + I or Q — p + 2, constants,
namely p — 1 or p constants,
if Q= 2p — 3, T+ 1 cannot be greater than 1, since the ratio of two of
the aggregates H (x) then vanishing at the poles would be a rational function
of the first order and therefore p be equal to unity — in which case 2p — 3 is
negative : thus the function involves p — 2 or p — 1 constants,
if Q = 2p — 4, and T + 1 be greater than unity, the ratio of two of the
vanishing aggregates fl (#) would be a rational function of the second order :
we have already several times referred to this possibility as indicative that
the surface is of a special character — called hyperelliptic — and depends in
fact only on 2p — 1 independent moduli. In general such a function would
involve p — 3 constants.
Ex. 2. Let V be an integral of the first kind and a be an arbitrary
definite place which is not among the 2p — 2 zeros of dV. We can form a
rational function infinite to the first order at the 2p — 2 zeros of dV and to
the second order at a; the general form of such a function would contain
2j9 — 2 + 2— p + I =p + 1 arbitrary constants. But there exists no rational
function infinite to the first order at the zeros of dV and to the first order at
46 IMPORTANT EXAMPLE. [37
the place a. Such a function would indeed by the Riemann-Roch theorem
here stated, contain 2p — 2 + 1— p -{• l=p arbitrary constants : but the coeffi
cients of these constants are in fact infinite only at the zeros of d V. For when
the places a1} ... , 0^-2 are all zeros of an aggregate of the form
AA(a;) + ...+Apnp(ac),
the conditions that the periods of an expression
be all zero, namely the equations
Xj nt (aO + . . . + \2p_2 fli (oap-a) + fjLfli (a) = 0, (i = 1, 2, . . . , p),
lead to
p, [AfMa) + ... + Apflp(d)] = 0,
and therefore to /JL — 0.
Thus the function in question will be a linear aggregate of p functions
whose poles are among the places a1} ... , a^-s- As a matter of fact, if W be
a general integral of the first kind, expressible therefore in the form
2F2 + ...+\PVP,
dW
wherein V2, ..., Vp are integrals of the first kind, v^ involves the right
a v
number of constants and is the function sought.
In this case the place a does not, in the sense of § 23, depend upon the
places a1} ... , 0^-2 j ^ne symbol suggested in § 26 for the places a1} ... , a^-a,
a, ... is
1,2,3, .. .,^-1,^+1, ...,2p-2, 2^-1, 2^,2^ + 1,....
It may be shewn quite similarly that there is no rational function having
simple poles in a1} a2, ... , a2p_2 and infinite besides at a like the single
term — , t being the infinitesimal at the place a.
v
Ex. 3. The most general rational function R which has the value c at
each of Q given distinct places, R — c being zero of the first order at each of
these places, is obviously derivable by the remark that l/(R — c) is infinite at
these places.
38]
CHAPTER IV.
SPECIFICATION OF A GENERAL FORM OF RIEMANN'S INTEGRALS.
38. IN the present chapter the problem of expressing the Riemann
integrals is reduced to the determination of certain fundamental rational
functions, called integral functions. The existence of these functions, and
their principal properties, is obtained from the descriptive point of view
natural to the Riemann theory.
It appears that these integral functions are intimately related to certain
functions, the differential-coefficients of the integrals of the first kind, of
which the ratios have been shewn (Chapter II. § 21) to be invariant for
birational transformations of the surface. It will appear, further, in the
next chapter, that when these integral functions are given, or,, more pre
cisely, when the equations which express their products, of pairs of them, in
terms of themselves, are given, we can deduce a form of equation to re
present the Riemann surface ; thus these functions may be regarded as
anterior to any special form of fundamental equation.
Conversely, when the surface is given by a particular form of fundamental
equation, the calculation of the algebraic forms of the integral functions may
be a problem of some length. A method by which it can be carried out is
given in Chapter V. (§§ 72 ff.). Compare § 50 of the present chapter.
It is convenient to explain beforehand the nature of the difficulty from which the
theory contained in §§ 38 — 44 of this chapter has arisen. Let the equation associated
with a given Riemann surface be written
wherein A, A1,..., An are integral polynomials in x. An integral function is one whose
poles all lie at the places .r=o> of the surface; in this chapter the integral functions
considered are all rational functions. If y be an integral function, the rational
symmetric functions of the n values of y corresponding to any value of .r, whose
values, given by the equation, are -AJA, Ay/A, -A^A, etc., will not become infinite
48 RATIONAL FUNCTIONS WHOSE POLES [38
for any finite value of x, and will, therefore, be integral polynomials in x. Thus when
y is an integral function, the polynomial A divides all the other polynomials Alt
A 2, ...... , An. Conversely, when A divides these other polynomials, the form of the
equation shews that y cannot become infinite for any finite value of x, and is therefore
an integral function.
When y is not an integral function, we can always find an integral polynomial in
x, say /3, vanishing to such an order at each of the finite poles of y, that /3y is an
integral function. Then also, of course, |32/2, /33y 3, . . . are integral functions: though it
often happens that there is a polynomial /32 of less order than /32, such that /32y2 *s
an integral function, and similarly an integral polynomial #3 of less order than /33,
such that ft3y* is an integral function ; and similarly for higher powers of y.
In particular, if in the equation given we put Ay = rj, the equation becomes
r,n + AlT)n-l + A2Ar,n-2 + ... + AnAn-l = 0,
and T) is an integral function.
Suppose that y is an integral function. Then any rational integral polynomial in
x and y is, clearly, also an integral function. But it does not follow, conversely,
though it is sometimes true, that every integral rational function can be written as an
integral polynomial in x and y. For instance on the surface associated with the
equation
f + Bfx + Cyx* + Dtf -E(f- A-2) = 0 ,
the three values of y at the places .r = 0 may be expressed by series of positive integral
powers of x of the respective forms
, y— -
Thus, the rational function (/• — Ey^x is not infinite when #=0. Since y is an
integral function, the function cannot be infinite for any other finite value of x.
Hence (y2 - Ey}jx is an integral function. And it is not possible, with the help of the
equation of the surface, to write the function as an integral polynomial in x and y.
For such a polynomial could, by the equation of the surface, be reduced to the form
of an integral polynomial in x and y of the second order in y ; and, in order that such
a polynomial should be equal to (y^-Ey^lx, the original equation would need to be
reducible.
Ex. Find the rational relation connecting x with the function 77 = (#2 — Ey}jx ; and
thus shew that 17 is an integral function.
39. We concern ourselves first of all with a method of expressing all
rational functions whose poles are only at the places where x has the same
finite value. For this value, say a, of x there may be several branch places :
the most general case is when there are k places specified by such equations as
x - a = £ri+1, • • • , x- a = tkwk+\
The orders of infinity, in these places, of the functions considered, will be
specified by integral negative powers of tlf . .., tk respectively. Let F be
such a function. Let o- + 1 be the least positive integer such that (x — aY+lF
is finite at every place x = a. We call <r + 1 the dimension of F. Let
f(xt y) = 0 be the equation of the surface. In order that there may be any
branch places at x = a, it is necessary that df/dy should be zero for this value
39] ARISE FOR THE SAME VALUE OF X. 49
of x. Since this is only true for a finite number of values of x, we shall suppose
that the value of x considered is one for which there are no branch places.
We prove that there are rational functions h1} ..., hn^ infinite only at
the n places x = a, such that every rational function whose infinities occur
only at these n places can be expressed in the form
(— > l] +(— > l) h+. ..+(-?— , l] hn. ..(A),
\ao -a J\ \x-a' A, \x - a' Jx^
in such a way that no term occurs in this expression which is of higher
dimension than the function to be expressed : namely, if a + 1 be the dimen
sion of the function to be expressed and o-; + 1 the dimension of hi, the
function can be expressed in such a way that no one of the integers
X, AX + al + 1, . . . , A,^ + a-n^ + 1
is greater than cr + 1. We may refer to this characteristic as the condition
of dimensions. It is clear conversely that every expression of the form (A)
will be a rational function infinite only for x = a.
Let the sheets of the surface at x = a be considered in some definite
order. A rational function which is infinite only at these n places may be
denoted by a symbol (R1} R2> ... , Rn), where R1} R2, ... , Rn are the orders of
infinity in the various sheets. We may call Rlf R2, ... , Rn the indices of the
function. Since the surface is unbranched at x — a, it is possible to find a
certain polynomial in - - , involving only positive integral powers of this
SO ^~ CL
1 \ 72
quantity, the highest power being [- -) , such that the function
\x — a i
l), = (£,$,, ...,,SU,0)say ......... (i),
'•"
— a.
is not infinite in the nth sheet at a; = a.
Consider then all rational functions, infinite only at x = a, of which the
nth index is zero. It is in general possible to construct a rational function
having prescribed values for the (n - 1) other indices, provided their sum be
p + 1. When this is not possible a function can be constructed* whose indices
have a less sum than p + 1, none of them being greater than the prescribed
values. Starting with a set of indices (p + 1, 0, ... , 0), consider how far the
first index can be reduced by increasing the 2nd, 3rd, ... , (n - l)th indices.
In constructing the successive functions with smaller first index, it will be
necessary, in the most general case, to increase some of the 2nd, 3rd, ...,
(n — l)th indices, and there will be a certain arbitrariness as to the way in
which this shall be done. But if we consider only those functions of which
the sum of the indices is less than p + 2, there will be only a finite number
* The proof is given in the preceding Chapter, (§§ 24, 28).
B. 4
50 SPECIFICATION [39
possible for which the first index has a given value. There will therefore
only be a finite number of functions of the kind considered*, for which the
further condition is satisfied that the first index is the least possible such that
it is not less than any of the others. Let this least value be r1} and suppose
there are ^ functions satisfying this condition. Call them the reduced
functions of the first class — and in general let any function whose nth index
is zero be said to be of the first class when its first index is greater or not
less than its other indices. In the same way reckon as functions of the
second class all those (with nth index zero) whose second index is greater
than the first index and greater than or equal to the following indices. Let
the functions whose second index has the least value consistently with this
condition be called the reduced functions of the second class ; let their
number be k2 and their second index be r2. In general, reckon to the ith
class (i < n) all those functions, with nth index zero, whose t'th index is
greater than the preceding indices and not less than the succeeding indices.
Let there be ki reduced functions of this class, with iih index equal to i\.
Clearly none of the integers t\, ... , rn_j are zero.
Let now (^ ... s;_! r{Si+1 ... sn_i 0),
where r{ >slt ... , i\ > st-_,, n > si+l, ... , n > sn-i,
be any definite one of the ki reduced functions of the iih class. Make a
similar selection from the reduced functions of every class. And let
($! . . . $£_! R{ Si+l . . . Sn-i 0)
be any function of the iih class other than a reduced function, so that
Ri > Si, ... , Ri> Si-i, Ri > Si+1 , . . . , Ri > Sn-i-
Then by choice of a proper constant coefficient X we can write
(& . . . £<_! Ri Si+i . . . Sn_! 0) - X (x - a)~(Ri~Ti) (sl . . . «;_! n si+l . . . sn_! 0)
in the form
(^...T^R/T^-.-Tn^Ri-ri) (ii),
where R{ < Ri', 2\ may be as great as the greater of S1} Ri — (n - s^, but is
certainly less than Ri] and similarly T2, ... , T^ are certainly less than Rt',
while T{+1 may be as great as the greater of $f+1, Ri — (rt — si+l), and is there
fore not greater than R^, and similarly Ti+2, ... , Tn^ are certainly not greater
than Ri.
* Functions which have the same indices are here regarded as identical. Of course the
general function with given indices may involve a certain number of arbitrary constants. By the
function of given indices is here meant any one such, chosen at pleasure, which really becomes
infinite in the specified way.
39]
OF A FUNDAMENTAL SYSTEM.
51
Further, if
, 1
be a suitable polynomial of order Ri — r\ in
1
\.x — a
(x — a)~l, we can write
\iv — a / tii-Vi
/<y a/ -p>> a' &' r\\ (\\\\
— (/o i ... io i—i £L i ij i+i ... ij n_i \}) V111^'
where R"i may be as great as the greater of R'{, R{ — rit but is certainly less
than Ri; S\ may be as great as the greater of 1\, Ri — r{, but is certainly less
than Ri; and similarly $'2, ..., $';_! are certainly less than R^; while S'i+l
may be as great as the greater of 7\-+1, Ri—ri, and is certainly not greater
than RI\ and similarly S'i+2, ... , S'n-\ are certainly not greater than Rt.
Hence there are two possibilities.
(1) Either (S\ . . . £'f_i R"i S'i+1 . . . £'n_i 0) is still of the ith class,
namely, R"i > Slf ... , R"i > S'i^ , R"i > S'i+1 , . . . , R"t > £'„_, ,
and in this case the greatest value occurring among its indices (R"i) is less
than the greatest value occurring in the indices of (Si... Si-i Ri Si+1 . . . $n_j 0).
(2) Or it is a function of another class, for which the greatest value
occurring among its indices may be smaller than or as great as Rt (though
not greater) ; but when this greatest value is Ri, it is not reached by any of
the first i indices.
If then, using a term already employed, the greatest value occurring
among the indices of any function (Ri, ..., Rn) be called the dimension of
the function, we can group the possibilities differently and say, either
(S\ . . . S'i^i R"i S'i+i . . . S'n-i 0) is of lower dimension than
(Si ... Si-i Ri Si+1 . . . Sn-i 0),
or it is of the same dimension and then belongs to a more advanced class,
that is, to an (i + &)th class where k > 0.
In the same way if (^ ... ^ r{ ti+i . . . tn-i 0) be any reduced function of
the t'th class other than (^ ... st-_i rf si+1 . . . sn^ 0), we can, by choice of a
suitable constant coefficient p,, write
(t-L ... ti—iTi t-+ ... t — 0) — /x (s ... s-_ r-s- s 0)
where r'i<ri, t\... £';_i may be respectively as great as the greater of the
pairs (ti, s^ ... (^_j, Si_i) but are each certainly less than rit while similarly
no one of t'i+1, ... , t'n-i is greater than rt.
The function (t\ ... t'^i r{ tft+1 ... £'n_i 0) cannot be of the ith class, since
no function of the tth class has its tth index less than rt : and though the
greatest value reached among its indices may be as great as rt (and not
greater), the number of indices reaching this value will be at least one less
4—2
52 EXAMPLE [39
than for (s1 . . . s;_j rt si+1 . . . sn^ 0). Namely (t\ . . . JV-i r'i t'i+l . . . t'n^ 0) is
certainly of more advanced class than (si . . . £;_! Vi Si+1 . . . sn^ 0), and not of
higher dimension than this.
Denote now by hlt ... , hn^ the selected reduced functions of the 1st,
2nd, ..., (n — l)th classes. Then, having regard to the equations given by
(ii), (iii), (iv), we can make the statement,
Any function (Sl... $;_j Rt Si+l ... $n_j 0) can be expressed as a sum of (I)
an integral polynomial in (x — a)~l, (2) one ofhly ... , An_j multiplied by such
a polynomial, (3) a function F which is either of lower dimension than the
function to be expressed or is of more advanced class.
In particular when the function to be expressed is of the (n — l)th class
the new function F will necessarily be of lower dimension than the function
to be expressed.
Hence by continuing the process as far as may be needful, every function
f=(S1... Si-! Ri Si+1 . . . Sn-i 0)
can be expressed in the form
(— , l] + (— , l) h*. .. + (—, l] hn^+F,, (v)
Ve-a A U-a' AI \ac-a' An_,
where F^ is of lower dimension thany!
Applying this statement and recalling that there are lower limits to the
dimensions of existent functions of the various classes, namely, those of the
&! + . . . + kn-! reduced functions, and noticing that the reduction formula (v)
can be applied to these reduced functions, we can, therefore, put every func
tion f=(S1... Si-! Ri Si+l . . . Sn-i 0) into a form
f— , l) + (— , l) hl+... + (— , l) hn-!.
\at-a J\ \x-a /\l \x-a /x^
Now it is to be noticed that in the equations (ii), (iii), (iv), upon which
this result is based, no terms are introduced which are of higher dimension
than the function which it is desired to express : and that the same remark
is applicable to equation (i).
Hence every function (R1} ... , Rn) can be written in the form (A) in such a
way that the condition of dimensions is satisfied.
40. In order to give an immediate example of the theory we may take
the case of a surface of four sheets, and assume that the places x = a are such
that no rational function exists, infinite only there, whose aggregate order of
infinity is less than p + 1. In that case the specification of the reduced
functions is an easy arithmetical problem. The reduced functions of the first
class are (m1} w2, m3, 0), where mx is to be as small as possible without being
smaller than m2 or w3 : by the hypothesis we may take
Wj + m3 + m3 = p + I.
40]
OF THE FUNDAMENTAL SYSTEM.
53
Those of the second class require m2 as small as possible subject to
ml + w2 + ra3 = p + 1, m2>ml, ra2 > w3 :
those of the third class require w3 greater than m1 and w2 but otherwise as
small as possible subject to n^ + m2 + ws = p + 1. We therefore immediately
obtain the reduced functions given in the 2nd, 3rd and 4th columns of the
following table. The dimension of any function of the t'th class being denoted
by <Ti + 1, the values of <rt- are given in the fifth column, and the sum
ar1 + a~2 + 0-3 in the sixth. The reason for the insertion of this value will
appear in the next Article.
P
Reduced functions of
the first class
Reduced functions of
the second class
Reduced functions of
the third class
°"l> ff-2> ff3
ff1 + ff.2 + ffx
= 3H-1
(M, M, M, 0)
(M-2, M+l, M + l, 0)
(M-l, M+1,M, 0)
(M, M+l, M-l, 0)
(M-l, M, M+l, 0)
M-1,M, M
3M-1
= 3N-2
(N,N,N-1,0)
(N,N-1,N,0)
(N-l, N, N, 0)
(N-I,N-1,N + 1,0)
N-1,N-1,N
3^-2
= 3P
(P+l, P, P, 0)
(P + 1,P + 1,P-1,0)
(P+1,P-1,P + 1,0)
(P-l, P + l, P + l, 0)
(P, P + l, P, 0)
(P, P, P + l, 0)
P, P, P
3P
Here the reduced functions of the various classes are written down in
random order. Denoting those first written by h1} h2, h3, we may exemplify
the way in which the others are expressible by them in two cases.
(a) When p = 3M — 1, we have, /* being such a constant as in equa
tion (iv) above (§ 39),
(M, M + 1, M- 1, 0) -fi(M- 2, M + 1, M + 1, 0)= {M, M, M + 1, 0},
the right hand denoting a function whose orders of infinity in the various
sheets are not higher than the indices given. If the order in the third sheet
be less than M + 1, the right hand must be a function of the first class and
therefore the order in the third sheet must be M. In that case, since a
general function of aggregate order p + 1 contains two arbitrary constants,
we have an expression of the form
(M, M + 1, M - 1, 0) = fjih, + Ah, + B,
for suitable values of the constants A, B.
If however there be no such reduction, we can choose a constant \ so
that
{M, M, M + I, 0} - \(M- 1, M, M+l, 0) = {M, M, M, 0} = A'h, -f
54 SUM OF DIMENSIONS OF FUNDAMENTAL SYSTEM
and thus obtain on the whole
(M,M+1,M- 1, 0) = fJis + \h3 + A'h, + B',
for suitable values of the constants A', B'.
(b) When p = 3P we obtain
(P + 1, P + 1, P - 1, 0) - \k, + A(P,P + 1,P,0) + B
= \h1 + A \fiht + Ch3 + D}+B.
Ex. 1. Shew for a surface of three sheets that we have the table
[40
p
*Ii h2
o-i. 0-2
(Tj + O-2
odd
/p + l p + i \ /p-i ^ + 3 \
p-1 p + l
V 2 ' 2 ' ) \ 2 ' 2 ' )
(p + 2 p \ fp p + 2 \
2 ' 2
\ 2 ' 2' / \2' 2 ' /
2' 2
*
2£c. 2. Shew, for a surface of n sheets, that if the places x = a be such that it is
impossible to construct a rational function, infinite only there, whose aggregate order of
infinity is less than _£> + !, a set of reduced functions is given by
kP.Jtr+l'-(k,..Jk,t-l,..^k-l,<)\(k-ltkt.JLtk-l+.^-l,Q) ...... (k-l,...,k- !,£,...£,(>)
/<r + 2.. A-i = (£- 1, ...,k-l,k + l, k, ...k, G)(k-I, ..., £-1, k,k + l, k, ...A, 0) ......
(k-l, ...,k-l,k, ...k,k+l, 0)
wherein p + I = (n—l)k — r (r<»— 1) and, in the first row, there are r numbers ^ — 1 in
each symbol, and, in the second row, there are r+l numbers k—\ in each symbol. In
each case k, ...k denotes a set of numbers all equal to k and £—1, ..., £—1 denotes a set of
numbers all equal to k — 1.
The values of crj, ..., cr,. + 1 are each k — l, those of o-,. + 2» •••> «"n-i are each ^- Hence
0-J+. .. +o> + ! + o-r + 2+. . .+<rn_1 = (r + !)(&- l) + (n-r- 2) A = (n-l)*-r- 1 =^?.
^!r. 3. Shew that the resulting set of reduced functions is effectively independent of
the order in which the sheets are supposed to be arranged at x=a.
41. For the case where rational functions exist, infinite only at the places
x = a, whose aggregate order of infinity is less than ^ + 1, the specification
of their indices is a matter of greater complexity.
But we can at once prove that the property already exemplified and
expressed by the equation o-1 + ... + <rn_^ = p, or by the statement that the sum
of the dimensions of the reduced functions is p + n — 1, is true in all cases.
For consider a rational function which is infinite to the rth order in each
sheet at x — a and not elsewhere : if r be taken great enough, such a function
necessarily exists and is an aggregate of nr — p + 1 terms, one of these being
an additive constant (Chapter III. § 37). By what has been proved, such a
function can be expressed in the form
- a
,
x - a
,,_,
42] EXPRESSED BY THE DEFICIENCY OF THE SURFACE. 55
where the dimensions of the several terms, namely the numbers
X, Xj + (T} + 1 , . . . , Xn_! + 0"n-l + 1 ,
are not greater than the dimension, r, of the function.
Conversely*, the most general expression of this form in which X^X^ ...,
Xn_! attain the upper limits prescribed by these conditions, is a function of the
desired kind.
But such general expression contains
(X + 1) + (Xj + 1) + ... + (Xn-, + 1),
that is (r + 1) + (r - O + . . . + (r - cr,^),
or nr — (a1+...+ o^) + 1
arbitrary constants.
Since this must be equal to nr —p+ 1 the result enunciated is proved.
The result is of considerable interest — when the forms of the functions hl...hn-l are
determined algebraically, we obtain the deficiency of the surface by finding the sum of the
dimensions of //x. . ,hn _ l . It is clear that a proof of the value of this sum can be obtained by
considerations already adopted to prove Weierstrass's gap theorem. That theorem and
the present result are in fact, here, both deduced from the same fact, namely, that the
number of periods of a normal integral of the second kind is p.
42. Consider now the places x = oo : let the character of the surface be
specified by k equations
_—fWi + l — fWk+l
— »l » • •• i — "k k >
X X
there being k branch places. A rational function g which is infinite only
at these places will be called an integral function. If its orders of infinity
at these places be respectively rlt r.2,..., rk and G [n-/(Wi+l)J be the least
positive integer greater than or equal to ^/(w; + 1), and p + 1 denote the
greatest of the k integers thus obtained, then it is clear that p + 1 is the
least positive integer such that or*^1' g is finite at every place x = oo . We
shall call p + 1 the dimension of g.
Of such integral functions there are n — 1 which we consider particularly,
namely, using the notation of the previous paragraph, the functions
(x - a)^+l hlt ,(x- a)°n-i+1 hn^ ,
which by the definitions of a-1} , o-n_! are all finite at the places x = a,
and are therefore infinite only for x = oo . Denote (x — a)0^"1"1 hi by </;. If hi
do not vanish at every place x = oo , it is clear that the dimension of <ft is
* It is clear that this statement could not be made if any of the indices of the function to be
expressed were less than the dimension of the function. For instance in the final equation of
§ 40 (a), unless /t, X, A' be specially chosen, the right hand represents a function with its third
index equal to
56 PARTICULAR CASE OF INTEGRAL FUNCTIONS. [42
o-j + 1. If however hi do so vanish, the dim'ension of gi may conceivably be
less than o-^ + l; denote it by pi 4 1, so that pi < a-^ Then x~(?i+v gi} and
therefore also (x — a)~(pi+l]gi) = (x — aYi~i>ihi, is finite at all places #=oo :
hence (# — a)'Y~pi /^ is a function which only becomes infinite at the places
x = a. But, in the phraseology of § 39, it is clearly a function of the same
class as hi, it does not become infinite in the nth sheet at x = a, and is of
less dimension than hi if a^ > p^ That such a function should exist is
contrary to the definition of hi. Hence, in fact, o\- = p^. The reader will
see that the same result is proved independently in the course of the present
paragraph.
Let now F denote any integral function of dimension p 4 1. Then
#-(P+I) F [s finite at all places x = oo : and therefore so also is (x — a)~(p+1} F.
This latter function is one of those which are infinite only at places x = a ; if
F do not vanish at all places x=a, the dimension cr + 1 of (x — a)~(p+1) F
will be p + 1 : in general we shall have a- < p.
By § 39 we can write
x-a /Al \x-a
where cr -f 1 > Xi + o-^ + 1,
and therefore, a fortiori,
p + 1 > \ + <Ti + 1 > \i + pi + 1.
Hence we can also write
F= (1, a; - a)x O - a)'-* + (!,«- a)Al (* - a)"-A^' & 4 ......
4 (1, a? - a)An-x (« - a)"-A»-r%-i ^^j,
or say
^=(1,^4(1,^,0! 4 ...... + (l,«U-i0n-i, ............ (B)
where /Ai4pi 4 1 =/) -cr. + p^4 1 =p + 1 -(^ - pf) <p + 1,
namely, there is no term on the right whose dimension is greater than that
of F (and each of /-i, p,lt ...... , fin_1 is a positive integer).
Hence the equation (B) is entirely analogous to the equation (A)
obtained previously for the expression of functions which are infinite only
at places x = a. The set (1, glf ...... , gn-i) will be called a fundamental set
for the expression of rational integral functions*.
It can be proved precisely as in the previous Article that p1 4 p2 4 ......
4 pn-\ = P- For this purpose it is only necessary to consider a function
* The idea, derived from arithmetic, of making the integral functions the basis of the theory
of all algebraic functions has been utilised by Dedekind and Weber, Theor. d. alg. Funct. e.
Verdnd. Crelle, t. 92. Kronecker, U. die Discrim. alg. Fctnen. Crelle, t. 91. Kronecker, Grundziige
e. arith. Theor. d. algebr. Grossen, Crelle, t. 92 (1882).
43] GENERAL PROPERTIES OF FUNDAMENTAL SYSTEMS. 57
which is infinite at the places #=oc respectively to orders r (Wj + 1), ...,
r (wk + 1). And the equations Sp = Scr = p, taken with <7f > pit suffice to shew
that a-i = pt. It can also be shewn that from the set gl . . . gn^ we can
conversely deduce a fundamental set 1, (x — 6)~(pi+1) <ft, ...,(x — b)~lpn-rl} gn-i
for the expression of functions infinite only at places x=b; these have the
same dimensions as 1, (ft, ..., gn-i*-
43. Having thus established the existence of fundamental systems for
integral rational functions, it is proper to refer to some characteristic pro
perties of all such systems.
(a) If Gl ... Gn-: be any set of rational integral functions such that
every rational integral function can be expressed in the form
(x, l\ + (x, l\ £x+ ...... + 0, 1)AB_1 Gn-, ............... (C),
there can exist no relations of the form
(X) iv + (*, IV, 0i + ...... + (x, i V^ £„_! = o.
For if k such relations hold, independent of one another, k of the functions
(TJ ... 6rn_i can be expressed linearly, with coefficients which are rational
in x, in terms of the other n — 1 — k. Hence also {3$, (32y2,. . . , (3n-i-k yn~l~k,
@n-kyn~k> which are integral functions when &,...,$„_* are proper poly
nomials in x, can be expressed linearly in terms of the n— 1— k linearly
independent functions occurring among Gi...Gn-i, with coefficients which
are rational in x. By elimination of these n — 1 — k functions we therefore
obtain an equation
A + A,y + ...... + An_kyn-k = 0,
whose coefficients A, Al} ...... , An-k are rational in x. Such an equation is
inconsistent with the hypothesis that the fundamental equation of the surface
is irreducible.
(6) Consider two places of the Riemann surface at which the inde
pendent variable, x, has the same value : suppose, first of all, that there
are no branch places for this value of x. Let X, \lt ...... , \n-i be constants.
Then the linear function
A. + Xj GI + ...... + \i-i Gn-i
cannot have the same value at these two places for all values of \,
For this would require that each of G1} ...... , Gn-\ has the same value
at these two places. Denote these values by a1} ...... , an_i respectively.
We can choose coefficients filt ...... , /zn_! such that the function
* The dimension of an integral function is employed by Hensel, Crelle, t. 105, 109, 111 ; Acta
Math. t. 18. The account here given is mainly suggested by Hensel's papers. For surfaces
of three sheets see also Baur, Math. Aniuil. t. 43 and Math. Annal. i. 46.
58 GENERAL PROPERTIES OF [43
which clearly vanishes at each of the two places in question, vanishes also
at the other n — 2 places arising for the same value of x. Denoting the
value of x by c, it follows, since there are no branch places for a; = c, that
the function
[l*i(Gi ~ ai) + ...... + Pn-i(Gn-i - a«_i)]/0 - c)
is not infinite at any of the places x = c. It is therefore an integral
rational function.
Now this is impossible. For then the function could be expressed in
the form
(x, 1)A + (x, 1)^ G, + ...... + (a?, !)„ GW_a ,
and it is contrary to what is proved under (a) that two expressions of
these forms should be equal to one another.
Hence the hypothesis that the function
A + A! GI + ...... + Xn_j 6rn_]
can have the same value in each of two places at which x has the same
value, is disproved.
If there be a branch place at x = c, at which two sheets wind, and no
other branch place for this value of x, it can be proved in a similar way,
that a linear function of the form
cannot vanish to the second order at the branch place, for all values of
A!, ...... , A7l_i namely, not all of G1} ...... , Gn-L can vanish to the second
order at the branch place. For then we could similarly find an integral
function expressible in the form
...... + pn-i £»-i)/0 - c).
More generally, whatever be the order of the branch place considered,
at x = c, and whatever other branch places may be present for x = c, it is
always true that, if all of Gly ...... , Gn-i vanish at the same place A of
the Riemann surface, they cannot all vanish at another place for which x
has the same value; and if A be a branch place, they cannot all vanish
at A t() the second order.
Ex. 1. Denoting the function
by K, and its values in the n sheets for the same value of x by K(l\ /if <-),..., K(n\ we
have shewn that, for a particular value of x, we can always choose X, X1)t.., Xn_1( so
that the equation K(l) = KW is not verified. Prove, similarly, that we can always
choose X, A!,..., Xn_x so that an equation of the form
) = 0,
where m1,..., mlc_1, mk are given constants whose sum is zero, is not verified.
43]
FUNDAMENTAL SYSTEMS.
59
Ex. 2. Let x = ylt...,yic be k distinct given values of x: then it is possible to
choose coefficients X, A!,..., p, Mi)"-) finite in number, such that the values of the
function
at the places x=y1, shall be all different, and also the values of the function, at the
places x=yz, shall be all different, and, also, the values of the function, for each of
the places #=y3,..., yt, shall be all different.
(c) If 1, HI, H2, , Hn-i be another fundamental set of integral
functions, with the same property as 1, Glt , Gn-\, we shall have
linear equations of the form
1 = 1
where a;, j is an integral polynomial in x.
Now in fact the determinant
For if I
j \ is a constant (i= 1, 2, ..., n — 1 ;
denote the value of Hi, for a general value
j = 1, 2, ..., n — 1).
of x, in the rth sheet of the surface, we clearly have the identity
1, 1,
,1
10 ,0
1, 1,.
1
' '•
. £,<">
/-» (i\ ri (2)
w n— i > "n— 1 >••••>
Gn-i(n}
ff (1) ff (2) ff («)
JJ n -i > •" n— i i ) L± 11—1
If we form the square of this equation, the general term of the square of
the left hand determinant, being of the form H^H^ + + Hi{n)Hj(n}, will
be a rational function of x which is infinite only for infinite values of x ; it
is therefore an integral polynomial in x. We shall therefore have a result
which we write in the form
TB_1) = V« . A (1, <?,, G,, ....... Gn^\
aitj \. A (1, H1} ...... , Hn^) may be called the
A (1, H,t ...... ,
where V is the determinant
discriminant of 1 , H^ , ...... , Hn^.
If /3 be such an integral polynomial in x that fty, = 77, say, is an integral
function, an equation of similar form exists when 1, tj, if, ...... , ijn~l are
written instead of 1, Hl} ...... , Hn^. Since then A (1, 77, rf, ...... , V1"1) does
not vanish for all values of x it follows that A (1, G1} G.2 ....... , G^n-i) does
not vanish for all values of x. (Cf. (a), of this Article.)
But because 1, Hlt H«, ...... , Hn_± are equally a set in terms of which all
integral functions are similarly expressible, it follows that A (1,H1, ...... ,Hn_^)
does not vanish for all values of x, and that
A (1, Glt ...... , G_1) = Vi2 A (1, H,, ...... , #„.,),
where V! is an integral function rationally expressible by x only.
60 FUNDAMENTAL SYSTEMS. [43
Hence V2 . Vt2 = 1 : thus each ofV and Vl is an absolute constant.
Hence also the discriminants A (1, Glt , Gn_^) of all sets in terms of
which integral functions are thus integrally expressible, are identical, save
for a constant factor.
Let A denote their common value and 771,..., rjn denote any n integral
functions whatever ; then if A fa, i)2, ..., rjn) denote the determinant which is
the square of the determinant whose (s, r)th element is T/'J1, we can prove, as
here, that there exists an equation of the form
A (%,%,..., *») = JfsAt
wherein M is an integral polynomial in x. The function A (77!, rj2,..., r)n) is
called the discriminant of the set 77!, tj2,..., rjn. Since this is divisible by A,
it follows, if, for shortness, we speak of 1, Hl,..., #„_,, equally with 77^
i}2>-"> *7n> as a set of n integral functions, that A is the highest divisor common
to the discriminants of all sets of n integral functions.
(d) The sets (1, GI, , Gn-i), (1, H1} , Hn^) are not supposed
subject to the condition that, in the expression of an integral function in
terms of them, no term shall occur of higher dimension than the function to
be expressed. If (1, gl} , gn-i) be a fundamental system for which this
condition is satisfied, the equation which expresses Gi in terms of 1, (ft,
g.2, , gn-i will not contain any of these latter which are of higher
dimension than that of G* Let the sets G1 , , Gn-! , g1 , , gn^ be each
arranged in the ascending order of their dimensions. Then the equations
which express Gly G2, , Gk in terms of gl, , gn_l must contain at least
k of the latter functions ; for if they contained any less number it would be
possible, by eliminating those of the latter functions which occur, to obtain
an equation connecting G1} , Gk of the form
(as, !)* + (*, l)Al 0,+ + (x, l\ 0* = 0;
this is contrary to what is proved under (a).
Hence the dimension of g^ is not greater than the dimension of Gk '•
hence the sum of the dimensions of Glf G2> , Gn-i is not less than the
sum of the dimensions of g1} g2, , gn-i- Hence, the least value which is
possible for the sum of the dimensions of a fundamental set (1, G1} , Gn-J
is that which is the sum of the dimensions for the set (1, <ft, , gn-i), namely,
the least value is p + n — 1.
We have given in the last Chapter a definition of p founded on
Weierstrass's gap theorem : in the property that the sum of the dimensions
of (ft,..., gn--i is p + n — 1 we have, as already remarked, another definition,
founded on the properties of integral rational functions.
Ex. 1. Prove that if (1, glt ..., gn^v\ (1, hlt ..., hn_l) be two fundamental sets both
having the property that, in the expression of integral functions in terms of them, no terms
44] THE COMPLEMENTARY FUNCTIONS. 61
occur of higher dimension than the function to be expressed, the dimensions of the
individual functions of one set are the same as those of the individual functions of the
other set, taken in proper order.
Ex. 2. Prove, for the surface
y«_
that the function
rt
satisfies the equation
rf - Crj2 + a2br) - «22ai = 0 >
and that
A (1, y, rj) = bW + lSa^bc - 27<Va22 - 4a1c3 - 4a263,
A(l, y, /) = a12 A(l, y, ij) A(l, 77, ^2) = «22A(1, y, r,) A (y, y\ r,) = a*<* A(l, y, ij).
In general 1, y, rj are a fundamental set for integral functions, in this case.
44. Let now (1, glt g.,, ...... , gn-\) be any set of integral functions in
terms of which any integral function can be expressed in the form
(x, 1)M + (x, 1 V, <7i + ...... + O, 1 ^ <7n_i ,
and let the sum of the dimensions of g1} ...... , #H_X be p + n — 1.
There will exist integral polynomials in x, (3lt /32, ...... ,/37l_i, such that
ftiy1 is an integral function: expressing this by glt ...... , gn-i in the form
above and solving for g^ ...... , gn-i we obtain* expressions of which the
most general form is
_ /*i, n-i
9i
where /*;,«_!, ...... > Pi,i, f*>i, Di are integral polynomials in x. Denote this
expression by gi (y, x}.
Let the equation of the surface, arranged so as to be an integral
polynomial in x and y, be written
f(y,x) = Q«yn + Qiyn-1+ ...... + Qn-i y + Qn = o,
and let ^ (y, x) denote the polynomial
Qo y*+ &3T1 + ...... + Q,--i y + Qt-,
so that ^0 (y, #) is Q0.
Let ^>0', 0i', ...... , ^'n_! be quantities determined by equating powers of y
in the identity
* Since JT,, ..., <;n_j are linearly independent.
62 ALTERNATIVE DEFINITIONS OF [44
in other words, if the equations expressing 1, y, y2, , yn-1 in terms of
1-1,
iin~l — n 4- n « 4- -I-/7 n
y ~ "'n— l T W'n— i, i </i r T <*n— i, n— l J/n— u
where the coefficient G^J is an integral polynomial in x divided by /:?;, then
r O /x^1 — 1 V«7 * s 1 s\,^ 2 \,7 > / r • • • • • • i **"n — 1 /^Q
So that if we write
n being the matrix of the transformation, we have
where %/ = %; (y', #), and H represents a transformation whose rows are the
columns of H, its columns being the rows of D.
But if (Q) denote the substitution
Qn-2, Qn-3, , Qo, 0
ft, ft, 0,
Q0, 0,
we have
Hence, changing y' to y in fa' and writing therefore fa for fa', we may write
Either this, or the original definition, which is equivalent to
y'-y
= %o y"-1 + y71-2 %i (y', «) + + y %n-2 (y', <*) + x«-i (y, *) (F),
may be used as the definition of the forms fa, fa, , <£n_j.
The latter form will now be further changed for the purposes of an
immediate application : let ylf , yn denote the values of y corresponding
44]
THE COMPLEMENTARY FUNCTIONS.
63
to any general value of x for which the values of y are distinct. Denote
fc (Vr, *), ffi (yr, *), by fc<", <7*(r)> etc.
Then putting in (F) in turn y = tf = y1 and y' = ylly = y», we obtain
= 2' 3>
Hence if, with arbitrary constant coefficients cfl, c1} , Cn_i, we write
Co<£o(1> + C^1" + + C,^ ^ = </>(1),
we have
' c0 Cj cn_! ^>(1) = 0,
1 n I1) rt I1' /
1 .<7i ' yn-i J
or
/'(*
1
1 ^1(1)
^n-i
1 ^
0
r 0
1 9^
Cji— i
ffn-i
(n)
.(G);
and we shall find this form very convenient: it clearly takes an inde
terminate form for some values of x.
If we put all of d, ...... , Cn-i, = 0 except cr, and put cr = 1, and multiply
both sides of this equation by the determinant which occurs on the left hand,
the right hand becomes
where, if sijj = gi^ #/> + g
in the determinant
+ ...... +^"1' g}™, Sitj means the minor of sitj
$1 *1, 1 ^1, 2
1, n— i
Sn— i Sn—i, i *n— i, 2 *n— i, n— i
Since this is true for every sheet, we therefore have
<f>r _ Sr + Srt ij(/i+ + Sr> n-i ffn-i
"
^aA !_ 3A 1 ^A_
64 INVERSE DETERMINATION [44
and therefore, also
The equation (H) has the remarkable property that it determines the
functions ,,( . from the functions gt with a knowledge of these latter only.
J \<y x
But we can also express g1} ...... , gn-i so that they are determined from
y , y , ...... , -FTJ\ , with a knowledge of these only.
For let these latter be denoted by 70, 71, ...... , yn-i' and, in analogy with
«
the definition of sr, i, let a-ft f = "2 <yr{s} 7t(s).
s=l
Then from equation (H)
n I T 1
S 7r(S) #<«> = X >SU + Sr, i «i, i + ...... + Sr, n-i Si, n-i
«=i ^ L
= 0 or 1 according as z =}= r or t = r.
Therefore, also, by equation (H),
+i ...... •
s=i
1
so that equation (H) may be written
Jr = <Tr, o + °V, 1 9\ + ...... + °V, n-i ^n-i-
If then Sr, i denote the minor of ov, < in the determinant of the quantities
<rr ti — which determinant we may call V (y0, <y1} ...... ,7*1-1) — we have, in
analogy with (H),
gr=^ (Sr 7o + Sril 7!+ ...... + Srin_!7»_i) ............... (K)*.
Of course V = -^ and 2r> i = -r- s.r> t-, and equation (K) is the same as (H').
Ex. 1. Verify that if the integral functions ffi, ..., gn-i have the forms
wherein Z)15 ...,!>„_! are integral polynomials in x, then <£0, ..., 0n_! are given by
* The equations (H) and (K) are given by Hensel. In his papers they arise immediately from
the method whereby the forms of >t , y2 , ...... are found.
45] EXPRESSION OF INTEGRAL OF THE THIRD KIND.
Ex. 2. Prove from the expressions here obtained that
65
and infer that 2 (dv/d.v)i = 0,
8=1
v being any integral of the first kind.
45. We are now in a position to express the Riemann integrals.
Let P£ £ be a general integral of the third kind, infinite only at the
places xlt scz. Writing, in the neighbourhood of xl, x — xl = tlWt+l, dP/dx
will (§§ 14, 16) be infinite like
namely, like
dP
y
thus (x — #1) ^ is finite at the place x1 and is there equal to
Similarly (x — xz) -,-- is finite at #2 and there equal to
w2+ 1'
Assume now, first of all, for the sake of simplicity, that at neither x = x±
nor x — x% are there any branch places ; let the finite branch places be at
At any one of these where, say, x = a + tw+1, dPjdx is infinite like
1 d
(w + l)f
dP .
-V +...],
and therefore (x — a) -=- is zero to the first order at the place.
7 dx
Hence, if a = (x — aa) (x — a,). . .
be the integral polynomial which vanishes at all the finite branch places of
the surface, and g be any integral function whatever, the function
K.a.g.(»-^)(,-^
is a rational function which is finite for all finite values of x and vanishes at
every finite branch place.
Therefore the sum of the values of K in the n sheets, for any value of x,
being a symmetrical function of the values of K belonging to that value of x,
is a rational function of x only, which is finite for finite values of x and is
therefore an integral polynomial in x. Since it vanishes for all the values of
66 EXPRESSION OF INTEGRAL OF [45
x which make the polynomial a zero, it is divisible by a, and may be written
in the form aJ.
Let the polynomial J be written in the form
Xx (x - a;2) - X2 (x - X-L) + (x - x^ (x - x2) H,
wherein 7^ and X2 are constants and H is an integral polynomial in x. This
is uniquely possible. Let H be of degree ^ - 1 in x ; denote it by (x, \Y~\
Then, on the whole,
(g = - - - -- + (.. I)-'.
—
Multiply this equation by a; — a^ and consider the case when x = xl} there
being by hypothesis no branch place at as = xt. Thus we obtain the value of
Xj ; namely, it is the value of g at the place x^ This we denote by g(xly y^.
Similarly X,, is g (ara, y2). Further, at an infinite place where as = t-(w+l),
dP = tw+^ dP
dx w + 1 dt
so that x^dPjdx is finite at all places x = oc . Hence if p + 1 be the dimen
sion of the integral function g, and we write
a-P-i (^ _ tfj) ^p-1 (a; - x.2)
we can infer, since p cannot be negative, that yu, is at most equal to p.
Hence, taking g in turn equal to 1, glt ..., gn-i, the dimensions of these
functions being denoted by 0, r, + 1, ... , rn-, + 1, we have the equations
/
V
dP\ dP\
. . + = -
1 (dx), yi \dx)n x-x, x-
(-}
\dx/i
where r\, ... , r'7l_] are positive integers not greater than Tlf ... , TW_I respectively.
Let these equations be solved for (-5-) : then in accordance with equa-
\dxj-i
tions (G) on page 63 we have, after removal of the suffix,
45] THE THIRD AND FIRST KIND. 67
f (y) = (x, IV''-1 <f>, + (x. 1)T'*~J <k + . . . + (x, IVVi"1 <f»rt_i
dx
+
vU ^^ *^\
where </>i stands for <£; (a;, y).
This, by the method of deduction, is the most general form which dP/dx
can have ; the coefficients in the polynomials (x, I)1"'*"1 are in number, at most,
T! + T2 + ... +TH_!,
or p ; and no other element of the expression is undetermined. Now the
most general form of dP/dx is known to be
1 dx p dx \dx J '
wherein f ~^- 1 is any special form of -y- having the necessary character, and
\i , ..., \p are arbitrary constants. Hence, by comparison of these forms, we
can infer the two results —
(i) The most general form of integral of the first kind is
f dx ,_j
J f(y} X''J ' 0n-i(^, y)J,
wherein r'i < T; and the coefficients in (x, I)7'"1 are arbitrary :
(ii) A special and actual form of integral of the third kind logarithmically
infinite at the two finite, ordinary, places (xly y^, (x», 7/2), namely like
log [(x — x1)/(x — x2)], and elsewhere finite, is
f i 77 I I nr o"
J I \y J I tv t*/j
0o (iC> y} + 0i (x, y} gl (x2, y2) + . . . + 0n_a (x, y) gn-\ (&-2> 2/2)!
r _ y
A ^2 J
or
fx dx /"*• , d r^>0 (x, y) + 0! (x, y) gl (£, tj) + ... + 0n_i (x, y) gn-\ (j£, ri}~\
In the actual way in which we have arranged the algebraic proof of this
result we have only considered values of the current variable x for which the
n sheets of the surface are distinct : the reader may verify that the result
is valid for all values of x, and can be deduced by means of the definitions
of the forms </>„, ..., <£n_j, which have been given, other than the equation
(G).
Ex. Apply the method to obtain the form of the general integral of the first kind only.
5—2
68 DEDUCTION OF INTEGRAL OF SECOND KIND. [45
We shall find it convenient sometimes to use a single symbol for the
expression
<f>0 (as, y) + (/>! (x,
and may denote it by (#, £). Then the result proved is that an elementary
integral of the third kind is given by
em
Px' = \ dx \(x, #1) — (x, a?.,)"].
xltx.2 Jc LV '
This integral can be rendered normal, that is, chosen so that its periods at
the p period loops of the first kind are zero, by the addition of a suitable
linear aggregate of the p integrals of the first kind.
Now it can be shewn, as in Chapter II. § 19, that if Ex' c denote an elemen
tary integral of the second kind, the function of (x, y) given by the differ
ence
,„. ..... «(,dj: ^n;-*?'. M**pt« --••.!«
wherein D% denotes a differentiation, is not infinite at (£, •»?). It follows from
the form of P*' °x , here, that this function does not depend upon (x2, y«).
Hence it is nowhere infinite, as a function of (x, y}. Therefore, if not inde
pendent of (x, y), it is an aggregate of integrals of the first kind. Thus we
infer that one form of an elementary integral of the second kind, which is
once algebraically infinite at an ordinary place (£, •»;), like — (as — ^)~1, is
given by
dx_ d^ ftp (x, y) + 0! (as, y) gl ( £ ??)+... + <f>n-i (
The direct deduction of the integral of the second kind when the infinity
is at a branch place, which is given below, § 47, will furnish another proof of
this result.
46. We proceed to obtain the form of an integral of the third kind when
one or both of its infinities (xly yj, (<KZ, y») are at finite branch places ; and
when there may be other branch places for x = xl or x = x2.
As before, let a be the integral polynomial vanishing at all the finite
branch places. The function
ga (x — ajj) (x — #2) dP[dx
will vanish at all the places x = xl : and though it may vanish at some of
these to more than the first order, it will vanish at (x1} y^) only to as high
order as (x — x^}. Hence the sum of the values of this function in the several
sheets for the same value of x is of the form aJ, where J is a polynomial in x
which does not vanish, in general, for x = x± or x = x.^.
46] INTEGRAL OF THIRD KIND WHEN INFINITIES ARE BRANCH PLACES. 69
Hence as before (§ 45) we can write
/ dP\ I dP\ Xx X.,
Iff T- 1+ ••• + U7 j ~ 1= - +(x,iyt-1.
\ ax /i \ tW /n x — xl x — x»
Multiply this equation by x — xl and consider the limiting form of the
resulting equation as (x, y) approaches to (x1, y^) : let w + 1 be the number of
sheets which wind at this place. Recalling that the limiting value of
(x — x^dPjdx is l/(w+l), we see that w+I terms of the left hand, corre
sponding to the w+ 1 sheets at the discontinuity of the integral, will take a
form
where e is a (?y + l)th root of unity. The limit of this when t = 0 is
9(xi> y\)l(w + 1); the corresponding terms of the left will therefore have
9(xi>y\) as limit. The other terms of the left hand will vanish.
Hence Xj = g(xlt y^), X2 = ^(^2, y2). The determination of the upper limit
for p and the rest of the deduction proceed exactly as before. Thus,
The expression already given for an integral of the third kind holds ivhether
(%i> yi), (#2, y-) be branch places or ordinary places.
If we denote the form of integral of the third kind thus determined by
•P^ * > the zero c being assigned arbitrarily, it follows, as in § 45, above, that
an elementary integral of the second kind, which is infinite at a branch
place #!, is given by
Now if we write t for tXl and #/ =xl + tw+1, the coefficient of dxff'(y) in the
integrand of the form here given for Px'fc is
Xi , Xi
<t>« + 01 • (ffi + tg,' +...)+... + (/>n-i • (gn-i + tg'n-i + • • •)
x - a? -
wherein ^>0, ..., </>„_, are functions of a-, y, and ^, .... r/,^, #/, f//, ... are
written for 5r](^1) y,), ... , gn_, (Xl, y,), Dg^x,, y}), Dg,(xl} y,), ... , respectively,
D denoting a differentiation in regard to t. Hence the ultimate form is
70 EXAMPLES. [46
That is, introducing £, tj, instead of xly ylt an elementary integral of the
second kind, infinite at a finite branch place (f, rj), is given by
da; 0! (as, y) g( (£, rf) + . . . + <£n_! (x, y) #'„_, (£, 77)
/'(y) f-f
where </i (£, 77), ... are the differential coefficients in regard to the infini
tesimal at the place. It has been shewn in (6) § 43 that these differential
coefficients cannot be all zero.
Sufficient indications for forming the integrals when the infinities are at
infinite places of the surface are given in the examples below (1, 2, 3, ...); in
fact, by a linear transformation of the independent variable of the surface we
are able to treat places at infinity as finite places.
Ex. 1. Shew that an integral of the third kind with infinities at (xly y^, (x.2t #2) can
also be written in the form
(a?, y) ffr (xl , ?/i) X2 - * 00 (x, y ) + 2X2T»- <ftr (x,
_
./'(y) #-#1 ^-^2
wherein X1 = (^-a)/(*'1-a), \2 = (x-a)/(x.2-a}, T,. + I is the dimension of gr, and a is any
arbitrary finite quantity.
It can in fact be immediately verified that the difference between this form and that
previously given is an integral of the first kind. Or the result may be obtained by con
sidering the surface with an independent variable £ = (x — a)~l and using the forms of § 39
of this chapter for the fundamental set for functions infinite only at places x — a. The
corresponding forms of the functions <j> are then obtainable by equations (H) § 44.
Ex. 2. Obtain, as in the previous and present Articles, corresponding forms for inte
grals of the second kind.
Ex. 3. Obtain the forms for integrals of the third and second kinds which have an
infinity at a place x= QO .
It is only necessary to find the limits of the results in Examples 1 and 2 as (x1, y-^)
approaches the prescribed place at infinity. It is clearly convenient to take a = 0.
Ex. 4. For a surface of the form
y* = x(x-a1) ...... (#-02P + i),
wherein a1} ..., a2p + 1 are finite and different from zero and from each other, we may* take
the fundamental set (1, g^) to be (1, y\ and so obtain (00, </>1)=:(fy, 1). Assuming this,
obtain the forms of all the integrals, for infinite and for finite positions of the infinities.
Ex. 5. In the case of Example 4 for which /> = !, the integral of Example 1, when a
is taken 0, is
fdx r^ y+aft^-'yt _ ^ y + *?Xj~*yi\
] y \-x x — xv x x — x2 J"
Putting xl = QO and yl — tnx^ + nxt + A + Bxv ~ 1 + . . ., this takes the form
_^ fdx
^ J y
, [
-4|
J
| z
y _ x
dx F
Imz +
V L x-Xz x
Chap. V. § 56.
47] DIRECT PROOF FOR INTEGRAL OF SECOND KIND. 71
Prove that this integral is infinite at one place x = cc like logf-J and is otherwise
infinite only at (.vz, y.2), namely like — log (x — .>;2), if (.i'2, yz) be not a branch place.
Ex. 6. Prove in Example 5 that the limit of
2 / ~ \~ ~ ~ I ~
j y L'* •* **i •* j
as (,i\, yj) approaches that place (ao , oo ) where y = mx* + njc+A + B/jc + ..., is
y
and that the expansion of this integral in the neighbourhood of this place is
A 1
and that it is otherwise finite. It is therefore an integral of the second kind with this
place as its infinity. The process by which the integral is obtained is an example of the
method followed in the present and the last Articles, for obtaining an elementary integral
of the second kind from an elementary integral of the third kind.
47. We give now a direct deduction of the integral of the second kind
whose infinity is at a finite place (£, 77) : we suppose that (w + 1) sheets of
the surface wind at this place, and find the integral which is there infinite
like an expression of the form
<"•! , -4.J . . Aw Aw+1
T"T4 h>+^£'
t being the infinitesimal at the place.
Firstly, let F be an integral which is infinite like the single term (x — ^)~l>
so that in the neighbourhood of the infinity its expansion has a form
F= ±
Forming as before the sum of the values of the functions g . (x — £)2 dFJdx in
the n sheets of the surface, g being any integral function, we obtain an
expression
Putting x = % we infer, since all terms on the left except those belonging to
the place (£, 77) vanish, that
Differentiating, and then putting a; = f, we obtain, from the terms on the left
belonging to the infinity,
the summation extending to (w + 1) terms.
72 DIRECT INVESTIGATION [47
Now
r rl ~l 1 (J
•- ; — T^-^T:. -r. \V+* (B+2Ct + ...)'}
dos
vanishes when t is zero : hence
Hence we can prove as before that, save for additive terms which are
integrals of the first kind, the integral which is infinite like (as — £)~J is
given by
ix Dw+l [<£0 + </>!#! (f,
This result is true whether (^, ?;) be a branch place or an ordinary place.
Consider now the integral, say E, which is infinite at (f, 77) like t~m, m
being a positive integer less than w + 1. At this place, therefore, (x — £) dEjdx
171
is infinite like --- — y . — . If, as before, we consider the sum of the n values
of the expression a . g . (x — £) dE/dx, wherein </ is any integral function and
a is the integral polynomial before used, which vanishes at all the finite
branch points of the surface, we shall obtain
To find X, let x approach to £. Then all the terms on the left, except
those for the w+1 sheets which wind at the infinity of E, vanish : for such a
non-vanishing term we have an expansion of the form
where D denotes, as usual, a differentiation in regard to the infinitesimal of
the surface at (|, 77), and g is written for g (£, 77). The sum of these w+1
expansions is
Now in fact every summation ^tr, being a sum of terms of the form
wherein e is a primitive (w + l)th root of unity, will be zero unless r be a
multiple of w+1. Thus the terms involving negative powers of t in the
48]
OF INTEGRAL OK SECOND KIND.
73
sum will vanish : those involving positive powers of t will vanish ultimately
when t — 0 ; and in fact A is zero, otherwise E would contain the logarithmic
term A log (x — £) when (x, y) is near to (£, 77). Hence on the whole
A. — —
m — 1
Then, proceeding as before, we obtain an expression of the integral in the
form,
1 fx dx 1
~ ^ i J
Thus, denoting the expression
n-\
<£„ 0, y) + 2 <f>r (x, y) gr (£, 77)
i
by <t>, an integral which is infinite like an expression
A
•"• i/i-i-i
is given by
t
, ,
h
x dx
<t>
*^^ I >P.
Of course the differentiations at the place (£, 77) must be understood in
the sense in which they arise in the work. If <£ (£, 77) be any function of
£, 77, D<f> (£, 77) means that we substitute in <f> (x, y), for x, % + tw+1, and for y,
an expression of the form 77 + P(t\ that we then differentiate this function of
t in regard to t, and afterwards regard t as evanescent.
Ex. 1. Obtain this result by repeated differentiation of the integral pf'e .
Ex. 2. Obtain by the formula the integral which is infinite like A/t + JB/t'2 in the
neighbourhood of (0, 0), the surface being f = x(x, 1)3. Verify that the integral obtained
actually has the property required.
48. The determinant A (1, glt ... , gn_^, of which the general element is
can be written in the form
, x~ri~1sl
, x~Ti
X~Tn-i
-1
., orTn-r
In this form the determinant factor is finite at every place x = oo : hence
also ar <*-*+»»> A (1, ffl, ... , #„_,) is finite (including zero) at infinity. Thus
8 h
OF i
TJNIVERSII
74 THE ESSENTIAL FACTOR OF THE DISCRIMINANT. [48
A (1, glt ... , gn-i}, which is an integral polynomial in x, is of not higher order
than 2?i — 2 + 2p in x.
But when the sheets of the surface for x = oc are separate, it is not of less
order ; it is in fact easy to shew that if for any value of x, x = a, there be
several branch places, at which respectively w1 + 1, w2+ 1, ... sheets wind, then
A (1, g1} ..., gn_j) contains the factor (x— a)wi+w*+-.
For, writing, in the neighbourhood of these places respectively,
x-a = t1w'+l, a? — a = «2W»+1, ...,
the determinant (§ 43)
(1) ^
. or, . • , gzii ,
of which A(l, glt ..., gn-i) is the square, can, for values of x very near to
x = a, be written in a form in which one row divides by tlt another row by
ti2. ..., another row by t^1, in which also another row divides by t2, another
row by t.?,..., and another row by t.2w>, and so on.
Thus this determinant has the factor ^wi(«'i+1) t£w*(w*+l) . . . , and hence
the square of this determinant has the factor (x — a)Wl (x — a)w<i
Therefore, when there are no branch places at infinity, A (1, glt ...,gn-i)
has at least an order 2w, = 2n+.2p — 2 (§ 6).
In that case then A (1, gly ..., gn-\) is exactly of order 2n + 2p — 2: and,
when all the branch places occur for different values of x, its zeros are the
branch places of the surface, each entering to its appropriate order.
When the surface is branched at infinity, choose a value x = a where
all the sheets are separate: and let gi = (x — a)Ti+IA;. Then by putting
£ = (# — a)"1 we can similarly prove that A(l, Aa, ...,/fn_[) is an integral
polynomial in £ of precisely the order 2n + 2p—2. But it is immediately
obvious that
Hence if the lowest power of £ in A (1, h1} ... , An_i) be f", A (1, g1} ... , gn~i)
is an integral polynomial of order 2n + 2p — 2 — s. In this case the zeros of
A (1, gly ... , gn-i), which arise for finite values of x, are the branch places,
each occurring to its appropriate order, provided all the branch places occur
for different values of x: and A (1, hi, ... , hn-i) vanishes for x=<x> to an
order expressing the number of branch places there.
Ex. 1. For the surface y*=3?(x- !)(#-«) there are two branch places at x=0, and
a branch place at each of the places #=1, x=a, where all the sheets wind. Thus
-2 = w=2. 1 + 3 + 3 = 8.
Chap. II. § 21.
49] RATIONAL FUNCTION WITH p + I POLES. 75
For thi.s surface fundamental integral functions are given by ffl=y, 9<i=y^lx, ff3=y3/x-
With these values, prove that A (1, fflt g^ #3)= -256.1'2 (.>;- I)3 (x-aj\ there being a factor
.i-'2 corresponding to the superimposed branch places at .r = 0, while the other factors are of
the same orders as the branch places corresponding to them.
Ex. 2. The surface y^ = x^(x— 1) is similar to that in the last example, but there is a
branch place at infinity at which the four sheets wind, so that, in the notation of thi.s
Article, s = 3. As in the last example 2n + 2p — 2 = 8, and 1, y, y^lx^flx are a fundamental
system of integral functions. Prove that, now, &(1, <Ji, g2, y3) is equal to -25tu>2(x--l):{,
its order in x being 2/i + 2p — 2 — s = 8 — 3 = 5.
49. In accordance with the previous Chapter* the most general rational
function having poles at p + 1 independent places, is of the form AF+B,
where F is a special function of this kind and A, B are arbitrary constants.
The function will therefore become quite definite if we prescribe the
coefficient of the infinite term at one of the p + 1 poles — the so-called residue
there — and also prescribe a zero of the function.
Limiting ourselves to the case where the p + 1 poles are finite ordinary
places of the surface, we proceed, now, to shew that the unique function thus
determined can be completely expressed in terms of the functions introduced
in this chapter. It will then be seen that we are in a position to express
any rational function whatever.
If the general integral of the third kind here obtained with unassigned
zero be denoted by P^ a , the current variables being now (z, s), instead of
(x, y), the infinities of the function being at x and a, the function
« = </>•> (z, s) + fa (z, s}gl(x,y} + ...... + </>„_, (z, s) gn^ (x, y)
dz z- x
<fto (z, s) + fa (z, s) g, + ...... + ftn~i (z, s) «_!
(z, s) (z, IX'-1^- + fal_1 (z, s) (z, l)\-rl,
wherein glt ..., gnr_l are written for the values of the functions gl (z,s), ...,
f/n-i (z, s) at the place denoted by a, contains p disposeable coefficients,
namely, those in the polynomials (z, \}^~l, , (z, l)Tn-i~l.
Let now cl} , cp denote p finite, ordinary places of the surface, the
values of z at these places being actually clf ..., cpt which are so situated that
the determinant
wherein fa(r} is the value of fa (z, s) at the place cr, does not vanish. That it
is always possible to choose such p places is clear : for if vlt , vp denote a
* Chap. III. § 37.
76 ACTUAL EXPRESSION OF RATIONAL [49
set of independent integrals of the first kind, the vanishing of A expresses
the condition that a rational function of the form
involving only p — 1 disposeable ratios \l:\2: ...... : \p, vanishes at each of
the places C1} ...... , cp.
Choose the p coefficients in the function f'(s)dP/dz, so that this function
vanishes at clt ...... , cp : and denote the function dP/dz, with these coeffi
cients, by ty (a, a; z,^, ...... , cp), so that A/'(s) \Jr (a, a ; z, cx ...... cp) is equal
to the determinant
[z, x\ - [z, a], $! (z, s\ £</>! (z, s), ..., z*-1 ^ (z, s), ..., zTn-rl fa^ (z, s)
where [^, *•] denotes the expression
<£o (z, s) + <fti (z, s) g1(a:,y) + ...+ $n-i (z, s) g! (a, y)
Z — X
Suppose now that (z, s) is a finite place, not a branch place, such that
none of the minors of the elements of the first row of this determinant
vanish. Consider -v|r (x, a ; z, Cj , ...... , cp) as a function of (x, y). It is
clearly a rational function ; and is in fact rationally expressed in terms of all
the quantities involved. It is infinite at each of the places z, cu c2, ...... , cp —
and in fact as x approaches z, the limit of (z — x) ty (x, a ; z, cl} ...... , c^,) is
the same as that of
0o (z, s} + ^<j)r (z, s) gr (ay/)
/'(*)
namely, unity (§ 44, F) : so that at x = z, ty is infinite like — (x — z}~1. And
at GI, . .., Cp it is similarly seen to be infinite to the first order.
To obtain its behaviour when x is at infinity, we notice that, by the
definition of the dimension of gi (x, y), the expression
(x, y) . .[I z zri~l 1
- -•+ -r~ '
xri J
z — x \_x x
which is of the form
Zri
T.+Z ~\
V+...
x2
is finite for infinite values of x. If then we add to the first column of the
determinant which expresses the value of A/' (s) -\Jr (x, a ; z, clf ..., cp), the
following multiples of the succeeding p columns
g^y) ffi (a, 6) frfoy) 9-2 (a, b) ,_ ,_, „ ,
~~ ---
49] FUNCTION WITH p -f 1 ARBITRARY POLES. 77
the determinant will contain only quantities which remain finite for infinite
values of x.
On the whole then, as the reader can now immediately see, we can
summarise the result as follows.
•ty (x, a ; z,clt ...... , cp) is a rational function of x, having only p + 1 poles,
each of the first order, namely z, cl , ...... , cp. It is infinite at z like — (x — z)~l
and it vanishes at x = a.
It is immediately seen that if a function of x of the form
..
which is so chosen that it is zero at all of ct, ..., cp except Cj and is unity at
Ci, be denoted by wi (x), then -Jr (x, a; z, cl ... c«) is infinite at CL like *** ^ .
x - Ci
Let now R (x, y) be a rational function of (x, y) with poles at the finite
ordinary places z1} z2, ..., ZQ: let its manner of infinity at z.-t be the same as
that of — \i{x — Zi)~\ Then the function
R (x, y)-\l^(x,a: zlt clt ...,cp)-...-\Q^(x, a; zv, clt ..., cp)
is a rational function of (x, y) which is only infinite at clf ..., cp. Since
however these latter places are independent*, no such function exists — nor
does there exist a rational function infinite only in places falling among
c1} ..., cp. Hence the function just formed is a constant; thus
R(x, y) = \l^(x, a; zl} c,, ..., cp) +...+ \Q^(x, a; ZQ, c,, ...,cp) + \.
Conversely an expression such as that on the right hand here will represent
a rational function having zlt ..., ZQ for poles, for all values of the coefficients
\i, ..., \Q, \, which satisfy the conditions necessary that this expression be
finite at each of cl} ..., cp; these conditions are expressed by the p equations
*i *>i (z,) + \,a>i (z2) +. ..+ Xy a)i (ZQ) = 0,
where i = 1, 2, ..., p.
When these conditions are independent the function contains therefore
Q-p+l
arbitrary constants— in accordance with the result previously enunciated
(Chapter III. § 37). The excess arising when these conditions are not inde
pendent is immediately seen to be also expressible in the same way as before.
We thus obtain the Riemann-Roch Theorem for the case under con
sideration.
The function -^ (x, a ; z, c,, ..., cp) will sometimes be called Weierstrass's
function. The modification in the expression of it which is necessary when
* In the sense employed Chapter III. § 23,
78 ALGEBRAICAL DEDUCTION OF THE RESULTS [49
some of its poles are branch points, will appear in a subsequent utilization
of the function (Chapter VII.*). The modification necessary when some of
these poles are at infinity is to be obtained, conformably with § 39 of the
present chapter by means of the transformation x = (% — m)~l, whereby the
place a; = oo becomes a finite place £ = m.
50. The theory contained in this Chapter can be developed in a different
order, on an algebraical basis.
Let the equation of the surface be put into such a form as
wherein alt ..., an are integral polynomials in x: so that y is an integral
function of x.
By algebraical methods only it can be shewn that a set of integral
functions glt ..., gn_^ exists having the property that every integral function
can be expressed by them in a form
(x, l)A + (#, 1)A[ #i+...+ (#, I)A»-I gn-i,
in such a way that no term occurs in the expression which is of higher
dimension than the function to be expressed; and that the sum of the
dimensions of gl} ..., gn-i is not less than n—l but is less than that of any
other set (1, hlt ..., hn-i), in terms of which all integral functions can be
expressed in such a form as
If the sum of the dimensions of gl} ..., gn_^ be then written in the form
p + n — 1, p is called the deficiency of the fundamental algebraic equation.
The expressions of the functions gl} g2, ..., gn-l being once obtained,
and the forms <£0, ^>15 ...,<£„_! thence deduced as in this Chapter, the integrals
of the first kind can be shewn, as in this Chapter or otherwise^, to have the
form
d® 17 i w -i • \ , _
/'(y)
wherein r\ < rlt etc., T; + 1 being the dimension of g{. Thus the number
of terms which enter is at most TJ + + rn_i or p. But it can in fact be
shewn algebraically that every one of these terms is an integral of the first
kind, namely, that an integral of the form
is everywhere finite^ provided 0 ^r ^Tt- — 1.
* The reader may, with advantage, consult the early parts (e.g. §§ 122, 130) of that chapter at
the present stage.
t Hensel, Crelle, 109.
+ For this we may use the definition (G) or the definition (H) (§ 44). The reader may
refer to Hensel, Crelle, 105, p. 336.
50] OF THIS CHAPTER. 79
Then the forms of the integrals of the second and third kind will follow
as in this Chapter: and an algebraic theory of the expression of rational
functions of given poles can be built up on the lines indicated in the
previous article (§ 49) of this Chapter. In this respect Chapter VII. may be
regarded as a continuation of the present Chapter.
A method for realising the expressions of glt ..., <7n_j for a given form of
fundamental equation is explained in Chapter V. (§ 73).
For Kronecker's determination of a fundamental set of integral functions,
for which however the sum of the dimensions is not necessarily so small as
p + n — 1, the reader may refer to the account given in Harkness and
Morley, Theory of Functions, p. 262. It is one of the points of interest of the
system here adopted that the method of obtaining them furnishes an algebraic
determination of the deficiency of the surface.
CHAPTER V.
ON CERTAIN FORMS OF THE FUNDAMENTAL EQUATION OF THE RlEMANN
SURFACE.
51. WE have already noticed that the Riemarm surface can be expressed
in many different ways, according to the rational functions used as variables.
In the present chapter we deal with three cases : the first, the hyperelliptic
case (§§ 51 — 59), is a special case, and is characterised by the existence of a
rational function of the second order ; the second, which we shall often
describe as that of Weierstrass's canonical surface (§§ 60 — 68), is a general
case obtained by choosing, as independent variables, two rational functions
whose poles are at one place of the surface : the third case referred to
(§§ 69 — 71) is also a general case, which may be regarded as a generalization
of the second case. It will be seen that both the second and third cases
involve ideas which are in close connexion with those of the previous chapter.
The chapter concludes with an account of a method for obtaining the funda
mental integral functions for any fundamental algebraic equation whatever
($| 73—79).
It may be stated for the guidance of the reader that the results obtained for the
second and third cases (§§ 60 — 71) are not a necessary preliminary to the theory of the
remainder of the book ; but they will be found to furnish useful examples of the actual
application of the theory.
52. We have seen that when p is greater than zero, no rational function
of the first order exists. We consider now the consequences of the hypothesis
of the existence of a rational function of the second order. Let £ denote
such a function ; let c be any constant and a, ft denote the two places where
£=c, so that (f — c)'1 is a rational function of the second order with poles
at a, /3. The places a, /3 cannot coincide for all values of c, because the
rational function d^/dx has only a finite number of zeros. We may therefore
regard a, /3 as distinct places, in general. The most general rational function
which has simple poles at a, J3 cannot contain more than two linearly entering
arbitrary constants. For if such a function be \ + \ifi + X.2/2 + • • • > ^-> ^-i>
being arbitrary constants, each of the functions f1} f.2, ... must be of the
second order at most and therefore actually of the second order : by choosing
the constants so that the sum of the residues at a is zero, we can therefore
53] THE HYPERELLIPTIC CASE. 81
obtain a function infinite only at ft, which is impossible*. Thus the most
general rational function having simple poles at a, ft is of the form
-^ (£ - c)"1 + B. Therefore, from the Riemann-Roch Theorem (Chapter III.,
§ 37), Q-q=p-(r + I), putting Q = 2, q = 1, we obtain £>-(T + !) = !;
namely, the number of linearly independent linear aggregates
ft (x) = Xxfl, («)+...+ Xpftp («),
which vanish in the two places a, /? is p - 1. Since a may be taken arbitrarily
and c determined from it, and p — I is the number of these linear aggregates
which vanish in an arbitrary place, we have therefore the result — When there
exists a function of the second order, every place a of the surface determines
another place ft: and the determination may be expressed by the statement
that every linearly independent linear aggregate ft (x) which vanishes in
one of these places vanishes necessarily in the other.
53. Conversely when there are two places a, ft in which p — 1 linearly
independent ft (x) aggregates vanish, there exists a rational function having
these two places for simple poles. To see this we may employ the formula
of § 37, putting Q = 2, r + l=p-l, and obtaining q=l. Or we may
repeat the argument upon which that result is founded, thus — Not every
one of ftj (x), . . . , ftp (x) can vanish at a ; let ft, (a) be other than zero. Since
p - 1 linearly independent ft (x) aggregates vanish in a, and, by hypothesis,
p - 1 linearly independent ft (x) aggregates vanish in both a and ft, it
follows that every ft (x) aggregate which vanishes in a vanishes also in ft.
Hence each of the p — 1 aggregates
ft2 (a) ft, (x) - ft, (a) ft2 (x), ...... , flp (a) flx (x) - ftx (a) ftp (a),
vanishes in ft, namely, we have the p — 1 equations
fti(«)ft1(^)-ft1(a)fti(/3) = 0, (i = 2, 3,...,p).
Therefore the function
has each of its periods zero. Thus it is a rational function whose poles are at
a and ft : and ft, (/3) cannot be zero since otherwise the function would be of
the first order.
Hence when there are two places at which p — 1 linearly independent
ft(#) aggregates vanish, there is an infinite number of pairs of places having
the same character. For any pair of places the relation is reciprocal, namely,
if the place a determine the place ft, a. is the place which is similarly
determined by ft : in other words, the surface has a reciprocal (1, 1) corre
spondence with itself. It can be shewn by such reasoning as is employed in
* By the equation Q - q =p -(T + 1), if q were 2, r + 1 would be p, or all linear aggregates Q(x)
would vanish in the same places, which is impossible (Chap. II. § 21).
B. C
82 THE HYPERELLIPTIC CASE. [53
Chap. I. (p. 5), that if (xl} y^), (x2, y2) be the values of the fundamental
variables of the surface at such a pair of places, each of #j , ^ is a rational
function of xz and ?/2, and that conversely x2, y2 are the same rational
functions of xl and y^
54. We proceed to obtain other consequences of the existence of a rational
function, g, of the second order. If the poles of £ do not fall at finite distinct
ordinary places of the surface, choose a function of the form (£ — c)"1, in
accordance with the explanation given, for which the poles are so situated.
Denote this function by 2. Then* the function dz/dx has 2.2 + 2p— 2 = 2p + 2
zeros at each of which z is finite. Denote their positions by xly x2, ..., ^+2-
If these are not all finite places we may, if we wish, suppose that, instead of
x, such a linear function of x is taken that each of xl} ... , x2p+2 becomes
a finite place. They are distinct places. For if the value of z at X{ be Cf,
z — d is there zero to the second order : that another place x-} should fall at
Xi would mean that z — c; is there zero to higher than the second order,
which is impossible because z is only of the second order. By the expla
nations previously given it follows that a linear aggregate H (#), which
vanishes at any one of these places x1} ... , xw+2, vanishes to the second order
there. Hence there is no linear aggregate II (x) vanishing at p or any
greater number of these places, for H (x) has only 2p — 2 zeros. The general
rational function which has infinities of the first order at the places xl,...t xp+r
will therefore f contain a number of q + 1 of constants given by p + r — q = p,
namely, will contain r + 1 constants. Such a function will therefore not
exist when r = 0. In order to prove that a function actually infinite in the
prescribed way does exist for all values of r greater than zero, it is sufficient,
in accordance with §§ 23 — 27 (Chap. III.), to shew that there exists no
rational function having xly x2, ... , #f for poles of the first order for any
value of i less than p + 1. Without stopping to prove this fact, which will
appear a posteriori, we shall suppose r chosen so that a function of the
prescribed character actually exists. For this it is certainly sufficient that r
be as great as p j. Denote the function by h, so that h has the form
\,\lt...,\r being arbitrary constants.
Let h, h' denote the values of h at the two places (x, y), (xr, y'\ where
z has the same value. Then to each value of z corresponds one and only one
value of h + h', or h + h' may be regarded as an uniform function of z : the
infinities of h + h' are clearly of finite order, so that h + ti is a rational
function of z. Consider now the function (z - Cj) (z - C2) . . . (z - cp+r) (h + h').
* Chap. I. § 6.
t Chap. III. § 37.
£ Chap. III. § 27. For the need of the considerations here introduced compare § 37 of
Chap. III.
55] DEDUCTION OF CANONICAL EQUATION. 83
Since h and h' are only infinite at places of the original surface at which
z is equal to one or other of c1( ..., cp+r, this function is only infinite for
infinite values of z. As it is a rational function of z, it must therefore be a
polynomial in z of order not greater than p + r. Hence we may write
h + k'= (Z, \}p+rl(z - Cj) ... (Z - Cp+r).
But here the left hand is only infinite to the first order, at most, at any
one of d, ..., Cp+r — and the denominator of the right hand is zero to the
second order at such a place. Hence the numerator of the right hand must
be zero at each of these places, and must therefore be divisible by the
denominator. Thus h + h' is an absolute constant, = 20 say. From the
equations
h =
we infer then that S; + S'f is also a constant, = 2(7; say : for h was chosen to
be the most general function of its assigned character and the coefficients
X, .... \r are arbitrary. Thence we obtain
G = \ + \C1 + ... +\Cr.
We can therefore put
so that s will be a function of the same general character as h, such however
that s + s' = 0 : in its expression the constants \i , . . . , \r are arbitrary, while
the constants Clf ..., Cr depend on the choice made for the functions
S ?
•^l» •••> ^r-
55. Consider now the two places a, a? at which z is infinite. Choose the
ratios \ : X2 : ... : \r so that s is zero to the (r — l)th order at a. This can
always be done, and will define s precisely save for a constant multiplier,
unless it is the case that when s is made to vanish to the (r — l)th order
at a, it vanishes, of itself, to a higher order. In order to provide for this
possibility, let us assume that s vanishes to the (r — I + &)th order at a.
Since s' = — s, s will also vanish to the (r — l+ &)th order at «'. There will
then be other p + r - 2 (r - 1 + k), or p - r + 2 - k, zeros of s. From the
manner of formation this number is certainly not negative. Consider now
the function
f=(z-cj)...(z-cp+r)s?.
At the places where z is infinite / is infinite of order p + r — 2 (r — 1 + k),
or p - r + 2 - 2k times. At the places, xlt ..., xp+r where s is infinite, it is
finite; each of the factors z - clt ..., z - cp+r is zero to the second order at
the place where it vanishes. Since s2 = - ss', f is a symmetrical function of
the values which s takes at the places where z has any prescribed value.
Hence, by such reasoning as is previously employed, it follows that the func-
6—2
84 CANONICAL EQUATION. [oo
tion f is a rational integral polynomial in z of order p — r + 2 — 2k. Denote
this polynomial by H. By consideration of the zeros of/ it follows that the
2 (p — r + 2 — 2k) zeros of the polynomial H are the zeros of s2 which do not
fall at a or a'. But since the sum of the values of s at the two places where
z has any prescribed value is zero, it follows that s is zero at each of the
places Xp+r+i, '••, #2p+2- For each of these is formed by a coalescence of two
places where z has the same value, and at each of them s is not infinite.
Hence the polynomial H must be divisible by (z — cp+r+1) ... (z — 0^+2).
Thus, as H is a polynomial of order p — r + 2 — 2k in z, p — r + 2 — 2k must
be at least equal to 2p + 2 — (p + ?•) or to p — r + 2. Hence k is zero, and
the value of H is determinate save for a constant multiplier. Supposing
this multiplier absorbed in s we may therefore write
(z-c1)...(z- cp+r) s* = (2- cp+r+1) ...(z- Cop+2) (A) ;
and s is determined uniquely by the conditions, (1) of being once infinite at
xly ..., xpJfr, (2) of being (?• — 1) times zero at each of the places a, a' where z
is infinite. Denote s, now, by sp+r, and denote the function h from which we
started, which was defined by the condition of being once infinite at each of
a?!, ..., Xp+r, by hp+r, and consider the function (z — cp+r)sp+r. This function
is once infinite at each of x1} ..., xp+r_l} it is zero to the first order at xp+r,
and it is r— 1 — 1, = r - 2 times zero at each of the places a, a' where z is
infinite. Hence the function
(z - cp+r) sp+r (A + AjZ + ...+ Ar_2 2r~2) + B,
wherein B, A, Aly ..., Ar_2 are arbitrary constants, has the property of being
once infinite at each of xl} ..., xp+r_lt and not elsewhere. It is then exactly
such a function as would be denoted, in the notation suggested, by hp+r-1}
and it contains the appropriate number of arbitrary constants — and we can
from it obtain a function sp+r_lt having the property of being once infinite at
each of x1} ..., xpJrr_^ and vanishing (r — 2) times at each of the places a, a'
where z is infinite.
Ex. 1. Determine sp + r_1 in accordance with this suggestion.
Ex. 2. Prove that hp + r is of the form sp + r (A + A ^ + . . . + A r _ ^ ~ !) + B.
Ex.*. Prove that Ap + , + 4 is of the form s^r(A+A,z+ +Ar + t_l2^^^ + R
(*-Vtr+i)*»(*-Vfcr+t)
Ex. 4. Shew that the square root /(*-cj>+r + 1)...(s-c2p + 2) ^ ^ interpreted as an
V (z~ci)-"(z~cp + r)
one-valued function on the original surface.
56. The functions, z, sp+r are defined as rational functions of the x, y
of the original surface. Conversely x, y are rational functions of z, sp+r.
For* we have found a rational irreducible equation (A) connecting z and
* See Chap. I. § 4.
56] FUNDAMENTAL INTEGRAL FUNCTIONS. 85
sp+r wherein the highest power of sp+r is the same as the order of z. Hence
this equation (A ) gives rise to a new surface, of two sheets, with branch places
at z = c1>..., Czp+2, whereon the original surface is rationally and reversibly
represented.
It is therefore of interest to obtain the forms of the fundamental integral
functions and the forms of the various Riemann integrals for this new surface.
It is clear that the function
(Z - C,) . . . (Z - Cp+r) Sp+r 0, l)*_i ,
where k is a positive integer, and (z, \)k-\ denotes any polynomial of order
k—l, is infinite only at the places a, a' where z is infinite, and in fact
to order p + r — (r— l) + k — l, = p + k: and that, therefore, by suitable choice
of the coefficients in another polynomial (z, l)p+fc, we can find a rational
function
(z - d) ... (z - cp+r) Sp+r (z, !)*_! + (z, l)p+k,
which is not infinite at a', and is infinite at a to any order, p + k, greater
than p. Now, of rational functions which are infinite only at a, there are p
orders for which the function does not exist*. Hence these must be the
orders 1, 2, ... , p.
Hence, of functions infinite only in one sheet at z = oo , on the surface
(Z - d) ... (Z - Cp+r) Szp+r = (z - Cp+r+i) ...(z- C^+a),
that of lowest order is a function of the form
which becomes infinite to the (p + l)th order. Hence by Chapter IV. § 39,
every rational function which becomes infinite only at the places z = oo , can
be expressed in the form
(z, 1)X-K*, 1V*7,
and if the dimension of the function, namely, the number which is the order
of its higher infinity at these places, be p + 1, X and fi are such that
p + 1 > X, p + l> ft +p + 1.
Therefore also, if er = (z — cx) . . . (z — cp+r) sp+r = t) — (z, 1 )p+1 , in which case
equation (A) may be replaced by the equation
<f*=(z- d) (z - c,) ...(z- c2p+2),
we have the result that all such functions can be also expressed in the form
(z, l)v + (.s, I), <r,
with
Chap. III. § 28.
86 EXAMPLES. [56
By means of this result, hitherto assumed, the forms for the various
integrals given Chapter II., § 17, Chapter IV., § 46, are immediately
obtainable by the methods of Chapter IV.
57. Or we can obtain the forms of the integrals of the first kind thus —
Let v be such an integral. Consider the rational function
, . , .dv
8p+r(z-c1)...(z-cp+r)fa.
It can only be infinite (1) where z is infinite (2) where dz = 0, that is at
the branch places of the (sp+r, z) surface. It is immediately seen that the
latter possibility does not arise. Where z is infinite the function is infinite
to the order p + 1 — 2, or p— 1. Hence it is an integral polynomial in z of
order p — l. Namely, the general integral of the first kind* is
/• (z, \)p-ldz
58. Ex. 1. A rational function hp_k, infinite only at the places where z = cl, ..., cp_t,
contains p-k-p + r + l +l = r+2-£ arbitrary constants, where T + ! is the number of
coefficients in a general polynomial (z, l)p-i which remain arbitrary after the prescription
that (z, !)„_! shall vanish at c1} ..., cp_t. Prove this: and infer that Ap, Ap_1,...do not
exist.
Ex. 2. It can be shewn as in § 57 that at any ordinary place of the surface
rational functions exist, infinite only there, of orders p + l, p + 2, ...: the gaps indicated by
Weierstrass's theorem (Chapter III. § 28) come therefore at the orders 1, 2, ...,p. At a
branch place, say at z = c, the gaps occur for the orders 1, 3, 5, ..., (2p- 1). For, all other
possible orders, which a rational function, infinite only there, can have, are expressible in
one of the forms 2(p-k), 2p + 2r+l, 2p + 2r, where k is a positive integer less than p, or
zero, and r is a positive integer: and we can immediately put down rational functions
infinite to these orders at the branch place z=c and nowhere else infinite. Prove in fact
that the following functions have the respective characters
fe *)?-* fa l)ro- + (g-c)fo l)p + r (z, l)p + r
wherein (z, !),,_*, (z, l)r, (z, l)p + r are polynomials of the orders indicated by their suffixes
with arbitrary coefficients.
Shew further that the most general Q(:c) aggregate which vanishes 2p-2k times at the
branch place contains k arbitrary coefficients: and infer that the expressions given
represent the most general functions of the prescribed character (see Chapter III. § 37).
Ex. 3. Prove for the surface
Ax* + Rvy + Cy* + Pa? + Qtfy+Rxy*
that the function
Cf. the forms quoted from Weierstrass. Forsyth, Theory of Functions, p. 456,
59] IRREMOVEABLE CONSTANTS OF THE SURFACE. 87
wherein X and /* are arbitrary constants, is of the second order. And that there are six
values of z for which the pairs of places at which z takes the same value, coincide, these
places of coincidence being zeros of the function
2 (A x* + Bxy + Cf) + Px3 + Q^y + Rxy* + Sy3.
Prove further that a rational function which is infinite at these six places is given by
_ 2 ( Ax* + Bxy + Cy*) + P'x3 + QWy + R'xy* + S'y3
~
for arbitrary values of the constants P', Q', R', S'.
This function is, therefore, such a function as has been here called hp + r : and since there
are six places at which dz is zero, p is equal to 2 and r equal to 4.
Prove that the sum of the values of h at the two places other than (0, 0) at which z has
the same value is constant and equal to 2.
We may then proceed as in the text and obtain the transformed surface in the simple
hyperelliptic form. But a simpler process in practice is to form the equation connecting
z and h. Writing k = h—\ and Z=xjy, prove that
P {(PZ3 + QZ* + RZ+ ,S02 - 4 ( AZ* + BZ+ C) (a^ + a^Z3 + a^ + a3Z+ a4)}
= {(P - P) Z3 + ((? - Q) Z* + (R'- R) Z+ (S1 - £)}*.
Hence, if the coefficient of k2 on the left be written (Z, 1)6, and we write
Y= [(P' - P) Z3 + (Q'-Q)Z* + (R' - R) Z+ (S' -
= [2 (A x* + Bxy + Cy*} + Px3 + Qa?y + Rxy*
we have
Y* = (Z, l)e,
which is the equation of the transformed surface. And, as remarked in the text, the
transformation is reversible ; verify in fact that #, y are given by
x=2Z(AZ2 + BZ+ C)/[ r- (PZ3 + QZ* + RZ+ £)],
y = 2 (AZ* + BZ+ <7)/[ Y- (PZ3 + QZ* + RZ+ S)].
Hence any theorem referred to one form of equation can be immediately transformed so
as to refer to the other form.
59. The equation
o-2 = (z - d) (z - c2) . . . (z - c2p+2)
by which, as we have shewn, any hyperelliptic surface can be represented,
contains 2^-1-2 constants, namely clt C2, . . . , c^+2. If we write z — (ox + b)/(x + c)
we introduce three new disposable constants ; by suitable choice of these
the equation of the surface can be reduced to a form in which there are only
2p — 1 parametric constants. For instance if we put
(Z - C,) (C8 - C2)/(> - C2) (Ca - d) = XJ(x - 1)
and then, further,
s=A<r(z- c3)-P-\
where the constant A is given by
A = (c, - c,)* (c, - c,YI(c, - c2)*"H (c, - c4)» (GS - c5)i . . . (c3 - (Vw)1,
88 EXAMPLES. [59
the equation becomes
s2 = x (x - 1) (oc - a,) (x - as) . . . (x - o^+j).
wherein
ar = (cz - c3) (cr - c^Kc, - c2) (c3 - cr\
and the right-hand side of the equation is now a polynomial of order 2p + 1
only. Of its branch places three are now at x=0, x=l, #=oo, and the
values of x for the others are the parametric constants upon which the
equation depends. It is quite clear that the transformation used gives s, x
as rational function of or, z. Thus
The hyperelliptic su?face depends on 2p — 1 moduli only. Among the
positions of the 3p — 3 branch places upon which a general surface depends
(Chapter I. § 7), there are, in this case, 3p -3-(2p -l)=p -2 relations.
Thus a surface for which p = 2 is hyperelliptic in all cases. There are in
fact (p—l)p(p + l) = Q places* for which we can construct a rational
function of order 2 infinite only at the place.
A surface for which p = I is also hyperelliptic — but it is more than this
(Chapter I. § 8), being susceptible of a reversible transformation into itself in
which an arbitrary parameter enters.
Ex. 1. On the surface of six sheets associated with the equation
y6 = x ( x — a) (x - 6)4
there are four branch places, one at (0, 0) where six sheets wind, and at (a, 0) where six
sheets wind, two at (b, 0) at each of which three sheets wind. These count f in all as
w = 6
Hence, by the formula
putting n = 6, we obtain p = 2.
Thus there exists a rational function £ of the second order, and the surface can be
reversibly transformed into the form »?2 = (£, l)(i. In fact the function
is infinite to the first order at each of the branch places (b, 0), (a, 0) and is not elsewhere
infinite.
To obtain the values of £ at the branch places of the new surface, we may express either
x or y in terms of £. Since there are two places at which £ takes any value, each of x and
y will be determined from £ by a quadratic equation— which may reduce to a simple
equation in particular cases. When £ has a value such that the corresponding two places
coincide, each of these quadratic equations will have a repeated root.
Now we have
(x-bf
-
Chap. III. § 31. f Forsyth, Theory of Functions, p. 349,
59] EXAMPLES OF HYPERELLIPTIC EQUATIONS. 89
Hence
y2(£G_1)_y£5(a_ 2&) _&(«-&) £4 = 0.
The condition then is
P(«-26)2 + 46(«-6)£4(|«-l) = 0, or |* [a2 (£« - 1 ) + (« - 26)2] = 0.
The factor
is equal to
[a2 {(x -Vf-x (x - a)} + (a - 26)2 x (x - a)]/# (# - a),
which is immediately seen to be the same as
[x (a - 26) + ab]/x (x - a)
or
{[x (a - 26) + ab] [x - 6]2/y3}2.
Thus this factor gives rise to the six places at which x= - ab/(a - 26). And if we put
T, = [x (a - 26) + ab] [a? -
we obtain
which is then the equation associated with the transformed surface.
Then, from the equation
^ - s = \x (a - 26) + ab]/[x - 6],
we obtain
which give the reverse transformation.
Ex. 2. Prove for the surface
y3=x (x-a) (# - 6)2 (x-cf
that jo = 2 and that the function
£=(x-b}(x-c)ly
is of the second order. Prove further that
[a£3 _ 6 _ cp + 4fa (£3 _ !) = {[a _ b _ c) y& + Mcx - abc]/x (x - a)}2
Hence shew that the surface can be transformed to
and that
# = [a2|3 + ai; + 26c - ab - ac]/[a|3 + r) + 6 + c - 2a],
y = 2£2 [be + a2 - a6 - ac] [a2^3 + a^ + 26c - ab - ac] / [«|3 + rj + b + c - 2a]2.
Ex. 3. In the following five cases shew that j» = 2, that £ is a function of the second
order, that in each case »/2 is either a quintic or a sextic polynomial in £, and obtain each
of x and y as rational functions of £ and 17 ;
(a) yw=x(x-aY(x-bf, £ = (x-a) (x - 6)//, r) = Ja.(x-aji(x-bY
08) f = x(x-a)*(x-b)\ £ = (*-a)(.*-6)//, r, = Ja . (x - a)* (x - b)*!y*>
(y) ?/> = x(x-a}(x-b}\ £ = C*-6)/y, ,, = [>(a-26) + a6][.r-&]2/y:J
(8) y» = .r2 (.r - a)3 (# - 6)3(.r - c)4, £ = A-(.r - a) (x - 6) (# - c)/y2, T, = c.v (x - a)2 (a? - 6)2 (x - c)/y3
(c) y4=a?(^-a)2(^-6)2(.r- c)3, £ = (x-d)(x-b)(x-cy*/ft r, = c(x-a)(x-b)(x-c)/xy.
90 WEIERSTRASS'S CANONICAL EQUATION. [59
Ex. 4. Shew that the surface
yn=(x-ai}n\..(x-ar}nr
can always be transformed to such form that nly ..., nr are positive integers whose sum is
divisible by n : and in that form determine the deficiency of the surface. Shew also that,
in that form, the only cases in which the deficiency is 2 are those given in Exs. 1, 2, 3.
Prove that the cases in which p = l are*
y6=x(x — af(x — b}3t y*=x(x-a)(x—b\
y*=*x(x-a)(x- 6)2, yz=x(x-a}(x- 6) (x - c).
The results here given have been derived, with alterations, from the dissertation,
E. Netto, De Transformatione Aequationis yn = R(x} (Berlin, 1870, G. Schade).
The equation
yn = (x-ai}ni...(x-ar}nr
is considered by Abel, (Eitvres Completes (Christiania, 1881), vol. i., pp. 188, etc.
It is to be noticed that in virtue of Chapter IV. we are now in a position, immediately
to put down the fundamental integrals for the surfaces considered in Examples 1, 2, 3.
60. Passing from the hyperelliptic case we resume now the considera
tion of the circumstances considered in Chapter III. §§ 28, 31 — 36.
Consider any place, c, of a Riemann surface : and consider rational
functions which are infinite only at this place : all such functions will be
denoted by symbols of the form gN, the suffix N denoting the order of infinity
of the function at the place.
Let ga be the function of the lowest existing order. The suffixes of all
other existing functions gN can be written in the form N = pa + i, where
i < a. Since there are only p orders for which functions of the prescribed
character do not exist, all the values i = 0, 1 , ...,(«—!) will arise. Let /^a + i
be the suffix of the function of lowest order whose order is congruent to i for
modulus a. We obtain thus a functions
ffa>>
Then, if gma+i be any other function that occurs, m cannot be less than /*,-,
and a constant A, can be chosen so that g-ma+i — ty SW+fi which is clearly
a rational function infinite only at c, is not infinite to the order ^a + i.
Thus we have an equation of the form
wherein pa +j is less than ma + i. Proceeding then similarly with g^+j, we
clearly reach an equation of the form
wherein the coefficients A, B, ..., K, whose number is a, are rational integral
polynomials in ga.
* Cf. Forsyth, p. 486. Briot and Bouquet, ThSorie des Fonct. Ellipt. (Paris, 1875), p. 390,
62] WEIERSTRASS'S CANONICAL EQUATION. 91
In particular, if gr be any rational function whatever of the gN functions,
we have equations
g, = Al + Btf^a^ + + K$*a _ !«+«-i
_ a+a_1 (ii).
61. If these equations, regarded as equations for obtaining #Mia+1,...,
g^ _ a+a-i in terms of ga and gr, be linearly independent, we can obtain, by
solving, such results as
g».a+i = Qi,i (ffr - AJ + Qifl (9r2 - A,) + . . . + Qf> «_, (g"'1 - A^),
wherein Q{il} ..., Qi,a-i are rational functions of ga, which are not necessarily
of integral form.
If however the equations be not linearly independent, there exist equations
of the form
or say
wherein Plf P2, ..., Pa_l5 P are integral rational polynomials in ga. Denote
the orders of these in ga by X1} X^, ..., Xrt-i> ^ respectively; here P denotes
the expression
P^ + P2A2 +...+ Pa-^a-! .
Then Pk gk is of order aX^ + rk at the place c of the surface. In order
that such an equation as (iii) may exist, the terms of highest infinity at
the place c must destroy one another: hence there must be such an
equation as
a\jc +rJc = a\K + rk',
and therefore
rfa = (Xjf — Xjfc)/(& — k').
Now k and k' are both less than a : this equation requires therefore that
r and a have a common divisor.
62. Take now r prime to a ; then it follows that the equations (ii) must
be linearly independent. And in that case each of g^a+i, •••> g* _ a+<i-i can
be expressed rationally in terms of ga and gr, the expression being integral
in gr but not necessarily so in ga.
Also by equation (i) it follows that every function infinite only at c is
rationally expressible by ga and gr: and in particular that there is an
equation of the form
Lfr + Ll9a-1 + ... + L^g, + La = 0 (iv),
92 ALL THE SHEETS WIND AT INFINITY. [62
wherein L, L1} ..., La are integral rational polynomials in ga, of which
however, since gr is only infinite when ga is infinite, L is an absolute
constant. It follows from the reasoning given that the equation (iv) is
irreducible, and therefore belongs to a new Riemann surface, wherein ga and
gr are independent and dependent variables. Further, any rational function
whatever on the original surface can be modified into a rational function
which is infinite only at the place c, by multiplication by an integral
polynomial in ga of the form (ga - Etf* (ga - Etf* ....... Hence any rational
function on the surface is expressible rationally by ga and gr. Hence the
surface represented by (iv) is a surface upon which the original surface can
be rationally and reversibly represented.
Since g~l is zero to order a at the place where ga is infinite, it is clear that
the new surface is one for which there is a branch place at infinity at which all
the sheets wind.
To every value of gr there belong r places of the old surface, at which gr
takes this value, and therefore also, in general*, r values of ga. Hence the
highest power of ga in equation (iv) is the rth, and this term does actually
enter. While, because ga only becomes infinite when gf is infinite, the
coefficient of the term gra is a constant (and not an integral polynomial in gr).
The equation (iv) is the generalization of that which is used in introducing what are
called Weierstrass's elliptic functions, namely of the equation
This equation is satisfied by writing g.i~^(u\ g^ = ^(u}: it is a known fact that the
poles of jp(«) are at one place (where w = 0). This is not true of the Jacobian function
snu.
63. It follows from equation (i) that the functions
form a fundamental set for the expression of rational functions infinite only
at the place c of the surface, that is, a fundamental set for the expression
of the integral rational functions of the surface (iv). And, defining the
dimension D of such an integral function F as the lowest positive integer
such that g~ F is finite at infinity on the surface (iv), in accordance with
Chap. IV., § 39, it is clear that in the expression of an integral function by
this fundamental system there arise no terms of higher dimension than the
function to be expressed : this fundamental set is therefore entirely such
an one as that used in Chapter IV. If k be the order of infinity of an
integral function F, at the single infinite place of the surface (iv), it is obvious
k
that the dimension of F is the least integer equal to or greater than - .
* That is, for an infinite number of values of gr.
64] POSSIBLE POSITION OF THE GAPS OF A RATIONAL FUNCTION. 93
64. We shall generally call the equation (iv) Weierstrass's canonical form;
a certain interest attaches to the tabulation of the possible forms which the
equation can have for different values of the deficiency p. It will be sufficient
here to obtain these forms for some of the lowest values of p ; it will be seen
that the method is an interesting application of Weierstrass's gap theorem.
Take the case p=4>, and consider rational functions which are only infinite
at a single place c of a surface which is of deficiency 4. Such functions do
not exist of all orders — there are four orders for which such functions do not
exist ; these four orders may be 1, 2, 3, 4, and this is the commonest case*,
or they may fall otherwise. We desire to specify all the possibilities : their
number is limited by the considerations —
(i) If functions of orders kl} kz, ... exist, say Fl, F*, ... , then there exists
a function of order n^ + n2k.2 + ... , where n1} n2, ... are any positive integers.
In fact F^F^... is such a function.
(ii) The number of non-existent functions must be 4.
(iii) The highest order of non-existent function cannot bef greater than
2p - 1 or 7.
It follows that a function of order 1 does not exist, and if a function of
order 2 exists then a function of order 3 does not exist ; for every positive
integer can be written as a sum of integral multiples of 2 and 3.
Consider then first the case when a function of order 2 exists. Write
down all positive integers up to 2p or 8. Draw| a bar at the top of the
numbers 2, 4, 6, 8 to indicate that all functions of these orders exist —
12345678 (a).
If then the functions of orders 5 or 7 existed there would need to be
a gap beyond 8, which is contrary to the consideration (iii) above. Hence
the non-existent orders are 1, 3, 5, 7. We have thus a verification of the
results obtained earlier in this chapter (§ 58, Ex. 2).
Consider next the possibility that a function of order 3 exists, there being
no function of order 2. If then a function of order 4 exists, the symbol
will be
12345678,
a function of order 6 being formed by the square of the function of order 3,
that of order 7 by the product of the functions of orders 3 and 4, and the
function of order 8 by the square of the function of order 4. Thus there
would need to be a gap beyond 8. Hence when a function of order 3 exists
* Chap. III. 31.
t Chap. III. § 34. Also Chap. III. § 27.
+ Cf. Chap. III. § 2G.
94
POSSIBLE POSITION OF THE GAPS OF A RATIONAL FUNCTION [64
there cannot be one of order 4. If however functions of orders 3 and 5
exist the symbol would be
12345678 (£),
the function of order 8 being formed by the product of the functions of orders
3 and 5. So far then as our conditions are concerned this symbol represents
a possibility. Another is represented by the symbol
12345678 (7).
In this case however the existent integral function of order 8 is not expressible
as an integral polynomial in the existent functions of orders 3 and 7.
When a function of order 3 exists there are no other possibilities ; other
wise more than 4 gaps would arise.
Consider next the possibility that the lowest order of existent function
is 4. Then possibilities are expressed by
1 2 3 4~5 "6 7 8 (S),
1 2 3 477 6 7~8 (e),
12345 6T~8 (£),
as is to be seen just as before.
Finally, there is the ordinary case when no function of order less than
5 exists, given by
1234 5~6 7 8
(77).
For these various cases let a denote the lowest order of existent function
and r the lowest next existent order prime to a. Then the results can be
summarised in the table
p=4 a
r
Gaps at
orders
Fundamental
system of orders
Dimensions of
functions of
fundamental
system
Sum of
these di
mensions
P+rt-i
\(n-\)(r-\)-p
a 2
9
1, 3, 5, 7
0, 9
0, 5
5
5
0
/3 3
5
1, 2, 4, 7
0, 5, 10
0, 2, 4
6
6
0
y 3
7
1, 2, 4, 5
0,7,8
0, 3, 3
6
6
2
8 4
5
1, 2, 3, 7
0, 5, 6, 11
0, 2, 2, 3
7
7
2
e 4
5
1, 2, 3, 6
0, 5, 7, 10
0, 2, 2, 3
7
7
2
f 4
7
1, 2, 3, 5
0, 6, 7, 9
0, 2, 2, 3
7
7
5
f, 5
6
1, 2, 3, 4
0, 6, 7, 8, 9
0, 2, 2, 2
8
8
6
65]
FOE THE CASES p = 3, p = 4.
95
That the seventh and eighth columns of this table should agree is in
accordance with Chapter IV., § 41. The significance of the last column is
explained in § 68 of this Chapter.
Similar tables can easily be constructed in the same way for the cases
p=l, 2, 3.
Ex. 1. Prove that for p = 3 the results are given by
p = 3
a
r
Gaps at
orders
Fundamental
system of orders
Dimensions of
functions of
fundamental
system
Sum of
these di
mensions
P+a-l
«i
•2
1
i, 3, r>
0,7
0,4
4
4
IB
3
4
1,2,5
0,4,8
0, 2, 3
5
5
7
3
5
1,2,4
0,5,7
0,2,3
5
5
8
4
5
1,2,3
0, 5, 6. 7
0, 2, 2, 2
6
6
Ex. 2. Prove that for /> = 5, 6, 7, 8, the possible cases in which the lowest existing
function is of the third order are those denoted by the symbols
p — 5
123456789 10
1 2 3 4 5 6~7 8 JTlO
12345678 9~10 11~12
1 2 3 4 5 6 7 8 9 10 11 12
1 2 3 4 5 6 7 8 9 10 11 12 13 14
= 7 J 1 2 3 4 5 6 7 8 9 10 11 12 13 14
.1 2 3 4 5 6 7 8 9 10 11 12 13 14
a 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
p = 8 ,123456789 10 TTT2 13 14 15 16
ll 2345678 916 11 12~ 13 14 HTT6
65. We have already stated (Chap. IV. § 38) that when the fundamental
set of integral functions are so far given that we know the relations expressing
their products in terms of themselves, the form of an equation to represent
the surface can be deduced. We give now two examples of how this may be
done : these examples will be sufficient to explain the general method.
Take first the case p = 4, a = 3, r = 7. Denote the corresponding func
tions by g3, g7. In accordance with § 60 preceding, all integral functions can
be expressed by means of g3 and two functions g7, g8 whose orders are respec
tively = 1 and 2 for modulus 3 : in particular there are equations of the form
9? = 9s (03, 1)2 + g7 (0s, 1)2 + (03, 1)4
9rf* = 9* (ff*. 1)8 + 9i (ff» l\ + (ft, 1)5
#82 =08 (</3> 1). +07 (03, 1)3 + (03, 1)S
96 FORMATION OF THE EQUATION. [65
wherein (g3, 1)2 denotes an integral polynomial in y3 of order 2 at most, the
upper limit for the suffix being determined by the condition that no terms
shall occur on the right of higher dimension than those on the left. Similarly
for the other polynomials occurring here on the right.
Instead of g7, g8 we may clearly use any functions g7 - (g3, 1),, gs - (g3y 1)2.
Choosing these polynomials to be those occurring on the right in the value of
ffrffs) we may write our equations
9* = «2#8 + &07 + Q4 , #82 = 7S08 + a$7 + «5 , 9$* = & (A),
where the Greek letters denote polynomials in g3 of the orders given by
their suffixes.
Multiplying the first and last equations by gs and g7 respectively, and
subtracting, we obtain
g7& = gs (0^8+  + o4)
and thence, since* 1, g7, g8 cannot be connected by an integral equation of
such form,
0272 + «4 = 0, fifeo, - /33 = 0, ar2<*5 + /3o/35 = 0,
from which, as 0.3 is not identically zero, — for then g7 would satisfy a quadratic
equation with rational functions of g3 as coefficients — we infer
«5 + &«3 = 0 (B).
Similarly from the last two equations (A) we have
7 + oB)
and thence
/8B- 05,03 = 0, 0^ + 05 = 0, 72^3 + oa«4 = 0,
so that, since «3 cannot be zero — as follows from the second of equations (A) —
we have
VM + a4 = 0 (C).
The equations (B) and (C) have been formed by the condition that the
equations (A) should lead to the same values for gfa and gfg1t however these
latter products be formed from equations (A). We desire to shew that, con
versely, these equations (B) and (C) are sufficient to ensure that any integral
polynomial in g7 and gs should have an unique value however it be formed
from the equations (A). Now any product of powers of g1 and g8 is of one of
the three forms g7, gf , g7g*K. In the first two cases it can be formed from
equations (A) in one way only. In the third case let us suppose it proved
that K has an unique value however it be derived from the equations (A);
* Chap. IV. § 43.
66] FORMATION OF THE EQUATION. 97
then to prove that g7g8 K has an unique value we require only to prove that
g7 . ge K = gs . g7 K. Let K be written in the form ggL + g7M + N. Then the
condition is that g7 (LgJ + Mg7g8 + Ngs) shall be equal to g8 (Lg7g% + Mgf + Ng7).
This requires only #7 • #82 = #» • #7#8 and g7 . g7ga = ga . g? : and it is by these
conditions that we have derived equations (B) and (C). Hence also g^^K
has an unique value.
Thus every rational integral polynomial in g7 and gs will, when the con
ditions (B), (C) are satisfied, have an unique value however it be formed from
equations (A).
The equations (B) and (C) are equivalent to a4=-a272, As = a2«s,
a5 = — a3/32, and lead to
Thence
or g73 - P2gf + cr2 y2g7 - a22«3 = 0,
which is the form of equation (iv) which belongs to the possibility under
consideration.
The expression of the fundamental set of integral functions 1, gr, g% in terms of g3 and
- is therefore
66. Take as another example the possibility e, § 64 above, where
a = 4, r = 5, the orders of non-existent functions being 1, 2, 3, 6. For a
fundamental system of integral functions we may take 1, gs, g*, g7.
We have then such an equation as
9*9-1 = g? (04, l)i + c#52 + <75(<74, l)i + (&, l)s
where c is a constant : let this be written in the form
gsg? = «i#7 + g* + frgs + «s,
the constant c being supposed absorbed in gf.
Write hs for g^ — ^ and h7 for g7 — Ji5 — A ~ %ai-
Then
Replacing now /*5, h7 by the notation gs, g1 and a3 + fliA + <*i2 by «3 we may
write
9*97 = «s, i/72 = ^3 + aa^s + a, (7S2 + frg, , g,3 =
B.
98 EXAMPLES. [66
Hence the condition g5 . gf = g5g7 . g7 requires
from which
and thence
0^3= — ^a^a, or if 0^ is not zero, y3=—(3-iy.2.
Substituting this value for ys and the value g7= a3lg5= ^yz/ga in the
expression for g&s we obtain
or
9* ~ Jiffs* ~ P*9* + Piytfs - *i722 = 0,
which is then a form of the equation (iv) corresponding to the possibility (e).
In this case the fundamental integral functions may be taken to be
It is true in general, as in these examples, that the terms of highest order
of infinity in the equation (iv) are the terms ga, gr. For there must be two
terms (at least) of the highest order of infinity which occurs ; and since r is
prime to a, two such terms as gag*, 9a9r cannot be of the same order of
infinity.
Ex. 1. Prove that for p = 3 the form of the equation of the surface in the case where
and shew that this is reducible to the form
yz +yx (x + a) + x* + c^-r3 + a.^ + a^c + a4 = 0,
x being of the form Ag3 + B, y of the form Cg^ + Dg3 + E, A, B, C, D, E being constants.
Thus the surface depends on 3p - 4 or 5 constants, at most.
Ex. 2. The reader who is acquainted with the theory of plane curves may prove that
the homogeneous equation of a quartic curve which has a point of osculation, can be put
into the form
By putting # = »;/£, # = «/£> tnis takes the form of the final equation of Example 1. Com
pare Chapter III. § 32.
Ex. 3. Prove that for jt) = 3, the form of the equation of the surface in the case where
Ex. 4. Denoting the left hand of equation (iv) by f(gr, ga], df/dgr by f'(ffr) and the
operator
67] RATIONAL FUNCTIONS NOT EXPRESSIBLE INTEGRALLY. 99
by Z>, prove that if gm be any rational function which is infinite only where gu and gr are
infinite, there exists an equation
X<>J»-igm + XlD*-*gm + + Xa_l9m = 0,
where X0, , J^-i are polynomials in ga.
67. We have already in Chapter IV. referred to the fact that an integral
function is not necessarily expressible integrally in terms of the coordinates
x, y by which the equation of the surface is expressed, even though y be an
integral function. The consideration of the Weierstrass canonical surface
suggests interesting examples of integral functions which are not expressible
integrally.
In order that an integral function g whose order is p should be expressible
as an integral polynomial in the coordinates ga, gr of the surface, in the form
it is necessary that there should be a term on the right hand whose order of
infinity is the same as that of the function ; we must therefore have an
equation of the form
fj, = ma + nr
wherein m, n are positive integers. Since a polynomial in ga and gr can be
reduced by the equation of the surface until the highest power of gr which
enters is less than a, we may suppose n less than a.
This equation is impossible for any value of //, of the form nr — ka. And
since herein k may be taken equal to any positive integer less than nr/a, the
number of integers of this form, with any value of n, is E(nr/a), or the
greatest integer contained in the fraction nr/a. Hence on the whole there
are
°2 E(nr/a)
n=l
orders of integral functions which are not expressible integrally by ga and gr.
Corresponding to any order which is not expressible in the form nr — ka,
which is therefore of the form nr + ma, we can assign an integrally expressible
integral function * namely gnrg™ : hence the p orders corresponding to which,
according to Weierstrass's gap theorem, no integral functions whatever exist,
must be among the excepted orders whose number we have proved to be
"Z E (nr/a) orf £ (a - 1) (r - 1).
n = l
Though it does not follow that every integral function whose order is of the form nr + ma
can be expressed wholly in integral form.
t If a right-angled triangle be constructed whose sides containing the right angle are
respectively a and ?•, and the interior of the triangle be ruled by lines parallel to the sides
7—2
100 NUMBER OF ORDERS OF RATIONAL FUNCTIONS [67
Hence the number of orders of actually existing integral functions which are
not expressible integrally is
In the table which we have given for p = 4 (§ G4) the existing integral
functions which are not expressible integrally are, for the case (7), of orders 8
and 11 ; for the case (8} of orders 6 and 11 ; for the case (e) of orders 7 and
11 ; for the case (f) of orders 6, 9, 10, 13, 17 ; for case (77) of orders 7, 8, 9, 13,
14, 19. The reader can easily assign the numbers for the cases in which
^ = 3.
Ex. 1. Prove that for the surface
9*+9*(9*-c)+9f,9s(9v l\+9*(9*> Va = °>
the function
ffj=ff&(ff&-c')/ff3
is an integral function which is not expressible as an integral polynomial in g3 and <75.
Ex. 2. Prove that for the surface
where a2 = o (gs - kj (g3 - £2),
82 = (ff3-ki)fi + l>i,
/! being of the first order in g3, and c, blt k±, k^ being constants, the two following functions
are integral functions not integrally expressible — •
#8 =g^ (9i + As)/a2 1 ffu ='9i (ffr + bi)l(ffa ~ *i)-
68. The number ^ (a — l)(r-l)—p is susceptible of another interpre
tation which is in close connexion with the last. Let the set of fundamental
integral functions for the Weierstrass canonical surface be denoted by
1, Glt G2,..., 6ra_!. From the equations whereby 1, gr, gr,..., gar are
expressed in terms of them we are able (Chapter IV., § 43) to deduce an
equation
wherein A(l, gr, ..., gv"1) is formed as a determinant whose (i, j)th element
is the sum of the values of g*r+J~2 at the a places of the surface where ga has
the same value, and is therefore an integral polynomial in ga, A(l, GI, . . . , Ga-^)
is formed as a determinant whose (i, j)th element is the sum of the values of
Gi^Gj-! for the same value of ga, which also is an integral polynomial in
containing the right angle, and at unit distances from these sides and each other, so describing
squares interior to the triangle, the number of angular points interior to the triangle is easily
seen to be S E (nr/a). On the other hand if the right-angled triangle be regarded as the half of
n=l
a rectangle whose diagonal is the hypotenuse of the right-angled triangle, and the ruled lines be
continued into the other half, it is easily seen that the total number of angular points of the
squares interior to the whole rectangle is (a- 1) (r- 1).
69] NOT EXPRESSIBLE INTEGRALLY. 101
ga, and V is a determinant whose elements are those integral polynomials in
ga which arise in the expressions of 1, gr, ..., g"~l in terms of 1, GI, ... , (ra_i.
The determinant A (1, gr, ... ,g"~l) is the square of the product of all the
differences of the values of gr which correspond to any value of ga. It
therefore vanishes, for finite values of ga, when and only when two of these are
equal. If the form of the equation of the surface be denoted by f(gr, ga) = 0,
this happens when, and only when, df/dgr=0. Now df/dgr is an integral
polynomial in ga and gr, of order a— 1 in the latter. Regarded as a rational
function on the surface it is only infinite when ga and gr are infinite. It
follows from the fact (§ 66), that gar is a term of the highest order of infinity
which enters in the polynomial f(gr, ga\ that df/dgr is infinite, at ga = oc ,
to an order r(a— 1). This is therefore the number of finite places on the
surface at which df/dgr vanishes. Hence we infer that the polynomial
A(l,$.,—»$£-1j is of degree r(a-l) in ga.
Since there is a branch place at infinity counting for (a — I) branch
places, the polynomial A(l, Glt...,Ga-i) is of order 2a + 2p - 2- (a - 1)
= a-l + 2pin0fl(§§48, 61).
Thus V is of order
i[r(a-l)-(a-
that is, of order
This interpretation of the degree of v is of interest when taken in connexion with the
theorem — Every integral function can be written in the form
(ffa, gr)l(ffa, 1),
the numerator being an integral polynomial in ga and gr, and the denominator being an
integral polynomial in ga. All the polynomials (ga, 1) thus occurring are divisors of the
polynomial y. See § 48 and § 88 Exx. ii, iii*.
When the factors of v are all simple we may therefore expect to be able to associate
each of them, as denominator, with an integral function which is not integrally expressible.
In this connexion some indications are given in a paper, Camb. Phil. Trans, xv. pp. 430, 436.
For Weierstrass's canonical surface see also a dissertation, De aequatione algebraica...in
quandam formam canonicam transformata. G. Valentin. Berlin, 1879. (A. Haack.)
Also Schottky, Crelle, 83. Conforme Abbildung. . .ebener Flachen.
69. The method which has been exemplified in §§ 65, 66 for the formation
of the general form of the equation of a surface when the fundamental set
of integral functions is given, is not limited to Weierstrass's canonical surface.
Take for instance any surface of three sheets, and let 1, g1} g^ be any set
* Cf. Harkness and Morley, Theory of Functions, p. 268, § 186.
102 INDICATION OF GENERALIZATION OF [69
of fundamental integral functions with the properties assigned in Chapter IV.
§ 42. Then there exist equations of the form
£i#2 = 7 + ft $1 + a 9*
wherein the Greek letters denote polynomials in the independent variable
of the surface, x, whose degrees are limited by the condition that no terms
occur on the right of higher dimensions than those on the left.
Thus the dimension of ft is not greater than that of g2 and the dimension
of a is not greater than that of g^ Hence we may use #1 — a, g2 — (3 instead
of g1 and g2 respectively, and so take the first equation in the form glg3 = y>
the form of the other equations being unaltered. As before, there are con
ditions that these equations should lead to unique values for every integral
polynomial in gl and </2, namely
#2 (71 + &9i + *i9*) = 9iV> 9i (72 + «2<7i + &SO = #27-
These lead to the equations
7=a1a2, 71 =-
and thence to
(v)
Since every rational function can be represented rationally by x and
gl and g2 = &\&>tlgi , it follows that every rational function can be represented
rationally by x and glt Hence the surface represented by the first of these
two final equations is one upon which the original surface is rationally and
reversibly represented. So also is the surface represented by the second of
these equations.
The fundamental integral functions are derived immediately from the
equation, being
Ex. 1. Prove that the integrals of the first kind for the surface
f(ffi> x)=ffi3-Piffi2+aiP2ffi -0^02 = 0
are given by
where rj + 1, r2 + l are the dimensions of g± and gz and /' (g^ =
Ex. 2. Prove that for the case quoted in Ex. i, § 40, Chapter IV, the form of the
equation is, (i) when p is odd = 2n- 1, say,
Sfn3-angn2 + an_1an + lffn-ain-lan + 2 = 0,
70] WEIERSTRASS'S CANONICAL EQUATION. 103
where an_l, an, an+1, n,l + 2 are polynomials in x of the orders indicated by their suffixes,
(ii) when j9 is even = 2?i — 2, say,
ffn3 - «nffn* + Pn*ngn ~ /3»27n = 0,
where on, /3tt, yn, 8n are polynomials in x of the nth order.
Ex. 3. Writing ffi = nly, the first of the equations (v) becomes
Q2=0. (A)
If the dimensions of gl and gz be rl + l, T2-|-l, find the degrees of the polynomials
GI} /31} a2, /32. And prove that if the positive quadrant of a plane of rectangular co
ordinates (x, y) be divided into squares whose sides are each 1 unit in length, and a convex
polygon be constructed whose angular points are determined from this equation (A), by
the rule that a term xry" in the equation determines the point (r, s) of the plane, then the
number of angular points of the squares which lie within this polygon is p.
70. In obtaining the equation
9? ~ &#!2 + «!&#! - «!2«2 - 0 (E)
we have spoken as if the original surface were of three sheets. It is im
portant to notice that this is not necessary.
Suppose our given surface to be any surface for which a rational function
of the third order, £, exists. Take c so that the poles of the function (£ — c)~l,
which is also a function of the third order, are distinct ordinary places of the
surface. So determined denote the function by x. Let alf cr2, «s denote these
poles. Then just as in § 39 of Chapter IV. it can be shewn that there exist
two rational functions g^ and g2, only infinite in ax and a.2, such that every
rational function which is infinite only in Oj, a2, a3 can be expressed in the
form
wherein y, a, /3 are integral polynomials in x whose degrees have certain
upper limits determined by the condition of dimensions.
And as before we can obtain the equation (E). Further, if F be any
rational function whatever and Alf A2, ... be the values of x at the places
other than ax, a2, a3 at which F becomes infinite, it is clearly possible to find
a polynomial K of the form (x - A^ (x - A^)n* . . . such that .KTonly becomes
infinite at al} a2, a3. Hence every rational function of the original surface
can be expressed rationally by x and glt
Thus as x, g^ are rational functions on the original surface, (E) represents
a new surface upon which our canonical surface is rationally and reversibly
represented. And it is as much the proper normal form for surfaces upon
which a rational function of the third order exists as is the equation
104 SURFACE OF FOUR SHEETS. [70
o-2 = (z, l)2p+2, previously derived, for the hyperelliptic surfaces upon which a
function of the second order exists.
Ex. Obtain the hyperelliptic equation in this way.
71. In the same way we can obtain a canonical form for surfaces upon
which a function of the fourth order exists. We can shew that there exist
three functions glt gz, g3 satisfying such equations as
+ k,
wherein the nine coefficients are integral polynomials in a rational function x,
which is of the fourth order; and that the surface is rationally and reversibly
representable upon a surface given by the equation
+ aj)sk2 + a-jbjcz + aj)3ki = 0.
Ex. These coefficients alt ..., fc3 satisfy certain relations; prove that the conditions
that Sra.ff3*=ffzffa.g3, gl . g32=ffiff3 • 9v ffiffs- ffa=ffaffa-ffi are that the following nine
polynomials should be divisible by a polynomial A, whose value is a]2b3— a3a161-ot2^i2 >
Herein ?i1 = a3 — c1? ^1 = a263 — ^x.
In fact if
the results of the division of these nine polynomials by A are respectively
«o> &5> C5> a4> 64> C4> f/(i> ^0» C6>
while
72. When the order of the independent function, denoted in §§ 69—71 by x, is known,
and the dimensions of the fundamental integral functions in regard thereto, the general
forms of the polynomial coefficients in the equations, whereby the products of pairs of
these integral functions are expressed as linear functions of themselves, can be written
down. And thence, if the necessary algebra (such as that indicated in the example of
§ 71), which serves to limit the forms of these polynomial coefficients, can be carried out, a
canonical form of the equation of the surface can be deduced.
But the converse process may arise : when we are given a form of the fundamental
equation associated with the surface, we may require to replace the given equation by one
in which the dependent variable is one of the set of fundamental integral functions. More
generally we may replace it by an equation in which the dependent variable is an integral
function of the form
OF 1
UNIVEI
V1
OF Till
_ CAl
74] DETERMINATION OF FUNDAMENTAL INTEGRAL FUNCTIONS. 105
This replacement possesses a high degree of interest (§ 88. Ex. iii). In either case
it is necessary to be able to calculate the fundamental integral functions.
73. We give now sufficient explanation to enable the reader to calculate the expression
of the fundamental integral functions for any given form of the fundamental equation
associated with the Riemann surface. This equation may* be taken in the form
#*+y"~lai + ~-+ytt»-i + a» = 0> (A)
«15 ..., an being integral polynomials in x ; thus y is an integral function of x (§ 38).
The n values of any rational function, 17, which arise for the same value of x, will be
denoted by i^1), ... , ij(") and called conjugate values ; their sum will be denoted by 2^. If
any of the possible rational expressions of 17 be $ (x, y)/^ (#, y\ $ and ty being integral
polynomials in x and y, and if in the expression of >j('),
we multiply numerator and denominator by the product of the n — l values conjugate to
^(.r,^1)), the denominator will become an integral symmetric function of y(l\ ...,y(n\ and
can therefore be expressed by means of the equation (A), as an integral polynomial in x ;
and the numerator will take a form which can be expressed as an integral polynomial in
x and yW. Hence the value of any rational function, on the surface associated with the
equation (A), can be expressed in the form
_
1=
A, ..., An_ly D denoting integral polynomials in x, with no common divisor.
Thus, to determine the expression of the fundamental integral functions, we may
enquire what modification this general form undergoes when TJ is an integral function.
74. In the first place the denominator D must be such that Dz is a factor of the
integral polynomial f A (1, ?/, ...,yn~1) ; so that D is capable only of a limited number of
forms. For let x — a be a factor of Z), repeated r times, and write
Ai = (x-aYBi^Ci, (t = 0,l,... ,(n-l))
wherein d is a polynomial of order less than r ; since J, ..., An_l have no common divisor
which divides D, not all of C, Clt ... , Cn_l can be divisible by x-a. Then the function
is an integral function, when 17 is an integral function, as appears from its first form of
expression. Denote it by f.
Suppose Ci not divisible by x-a. From the equation f
Mi^-.^-U^+S..^-'^^!,^,...,^^,
recalling the form of the determinant which is the square root of the left hand side, we
infer
(^VA(1'<y'><M'yi~1'^)y<+1'-"'/l~1)=Vi2A(1'5ri'>>i'5r»-i)-
Hence, save for sign,
*/?!-(*- aX/Ct,
so that (x — aY divides v-
Thus the first step in the determination of the integral functions is to put A (!,_>/,
....y"-1) into the form MI*» ...u**, wherein MI( ... , ur are polynomials having only simple
* Chap. IV. § 38. f Chap. IV. § 43.
106 ACTUAL ALGEBRAICAL DETERMINATION OF [74
factors. This can always be done by the rational process of finding the highest divisor
common to A(l,y, ...,yn~1) and its differential coefficients in regard to x. It will include
most cases of practical application if we further suppose all the linear factors of
A(l>ty, ...,yn~1) to be known*.
75. Suppose then that x — a is a factor which occurs to at least the second order in
A(l,y, ...,<yn"1). Denote x-a by u. By the solution of a system of linear equations,
we can (below, § 78) find all the existing linearly independent expressions of the form
(a + a1y+...+an_lyn-1)lu,
wherein a, at , . . . , a,t _ 1 are constants, which represent integral functions. If the highest
power of y actually entering be the same in two of these integral functions, say in f and f ',
we can use instead of f a function of the form f — /if, where /i is a certain constant. By
continued application of this method of reduction we obtain, suppose, k integral functions,
of the form
£r = (a' + a\y+...+a'ryr)/U, (C)
wherein, since these functions are linearly independent, k is less than n, and the vahies of
r that occur are all different. These values of r that occur are among the sequence
1, 2, ..., (n— 1) ; let s denote in turn all the n— 1 — k other integers in this sequence. Put
£, for y*. Consider now the set of integral functions
*» lit •••>£*-!•
As before we can determine by the solution of a system of linear equations all the
linearly independent functions of the form
wherein /3,#i, ..., /3n_x are constants, which are integral functions ; and, as before, we can
choose them so that the f 's of highest suffix which occur shall not be the same in any two
of these integral functions. Then in place of 1, f1} ..., fn_1 we obtain a set 1, £15 ...,£„_!,
wherein gr is fr unless there be an integral function of the form
O' + /3'lCl + ...+/3'rtr)/«, (D)
wherein the f of highest suffix occurring is £r, in which case £r denotes this function.
Then we enquire whether there are any integral functions of the form
•y, ...,yn-i being constants. If there are, the process is to be continued t- If there are
none, let v denote any other linear factor occurring in A (1, y,..., yn~1} to at least the
second order. Then, as for the set 1, y, ..., y""1, we investigate what linearly independent
integral functions exist of the form
and continue the process for v as for u : and afterwards for all other repeated factors of
Aa,^...,^-1)-
76. When these processes are completed, we shall obtain a set of integral functions
!> »?]> •••) '/n-l)
such that there exists no integral function of the form
* In the work below, if u be a polynomial of order r, it is necessary to suppose a, a5 , ..., a* to
be polynomials of order ?•— 1.
+ The number of steps is finite, by § 74.
77] FUNDAMENTAL INTEGRAL FUNCTIONS. 107
wherein a, ..., an_t are constants, for any value of c. It is obvious now from the successive
definitions (C), (D), ... of the sets (1, ft, ..., fn-i)» (1, £1, ...,£n-i), ..., (1, ijlt ..., •;»_,), that
every power of y can be represented in the form
wherein v, vv, ..., vn_l are integral polynomials in x. Hence every integral function can
be written in the form
r, = ( A'+ El r,, + . . . + En _ l r,n _ J/F,
wherein E, ..., £!n-u F are integral polynomials in x without common divisor. If now
x— c be a factor of F and we write
Ei = (x-c) (fi + ai, i=0, 1, 2, ..., (n- 1),
at being a constant, the function
is an integral function, as appears from the form of the left-hand side. By the property
of the set 1, Vn •••> 'Jn-i there is no integral function having the form of the right-hand
side, unless each of a, alt ..., an_T be zero.
Hence each of E, ...,En _l are divisible by x — c. By successive steps of this kind it
can be shewn that every integral function can be written in the form
1Tln-l, (E)
wherein H, fll, ..., Hn-i are integral polynomials in x.
77. But in order that the set 1, rjl, ..., r}n_l should be such a fundamental set as
I>ffi> •••>.9rn-i> used in Chap. IV., there must be no terms occurring on the right-hand side
here, which are of higher dimension than rj. We prove now that this requires a further
reduction in the forms of 1, ^ ..., rjn_l, which is of a kind precisely analogous to the
reductions already described.
Let a + 1 be the dimension of 17, pf the order, and therefore also the dimension of the
polynomial ZT( (§ 76) and a-i + l the dimension of ^; we suppose o^ ^> o-2 ;j> ... :j> o-n_1 ;
then
Putting #=!/£, h=rtlx<T , hi = rnlxT , Hix~Pi=(\^)pit an integral polynomial in £,
this equation is
If now in equation (E) a term arises of higher dimension than rj, one of the integers
p — (o- + l), ..., pi + <Ti — O-,...
is greater than zero. In that case let r+l be the greatest of these integers. Then we can
write
^=(...+ (i,£Mi+. ..)/£,
wherein the symbols (1, £)mi denote integral polynomials in £. Putting
(liflm^A'i + Oi, (1 = 0, 1, 2, ..., w-1),
wherein a; is a constant, we have
Herein the left hand is a function which is not infinite when x is infinite. Hence,
108 ACTUAL ALGEBRAICAL DETERMINATION OF [77
when the set 1, r)l, ...,rjn^l are such that the condition of dimensions* is not satisfied,
there exist functions of the form
i.e. of the form
wherein a, ..., an_! are constants which are not infinite when £ is zero or x is infinite.
In virtue of their definition the functions hlt ...,/;„_! are not infinite when x is infinite,
and are therefore infinite only when x is zero or £ infinite. We may therefore regard them
as integral functions of £. And since there exists no integral function of the form rjifx, the
dimensions of klt ...,hn_l as functions of £ are o-j + 1, ..., o-n_1 + l.
As before determine a set of linearly independent functions of the form
a, ..., an_j being constants, which are not infinite when £ = 0, choosing them so that the h
of highest suffix which occurs is not the same in any two of the functions. Let the
function wherein the h of highest suffix is hr be denoted by &,., so that kr is of the form
kr =
Then
^ = ^+
is a function which is not infinite when #=0, as appears from the form of the right-hand
side ; it is therefore an integral function of x, and since kr is not infinite when x is infinite
it is an integral function of x whose dimension is only o>. Denote it by Qr. Then r)r can
be expressed in the form
r OY~f~l i OV — O"i , (Tf — OV- 1 /-v -\ /Tl\
T)r= -- [fl* +Wl* l + ...+fi.r-^r-l« ~ Gr], (*)
P-r
and in the right hand no term occurs of higher dimension than that of i;r, while Gr is of
less dimension than r)r. If then there be m functions such as kr, m of the functions
i7u ..., ijn-i can be expressed in the form (F) in terms of the remaining n — l — m functions
of ijj, ...,»;„_! and m functions Gr ; the sum of the dimensions of these m functions Gr is
less by m than that of the dimensions of the functions rjr which they replace. Denoting
the functions among i^, ..., tjn_l which are not thus replaced by functions G, also by the
symbol G, for the sake of uniformity, every integral function is expressible in the form
(x, l)A + (.r, l)^Gl + ... + (x, l^ffn-i,
and the sum of the dimensions of Gl, ..., Gn^l is less by m than the sum of the dimensions
°f »?i, •••j'Jn-i-
If now in this expression of integral functions by Gly ..., Gn_1 any terms can arise
which are of higher dimension than the functions to be expressed, we can similarly replace
the set G!, ..., Gn_l by another set whose dimensions have a still less sum.
Since no integral function can have a less dimension than 1, the sum of the dimensions
of the functions whereby integral functions are expressed, cannot be diminished below n — 1.
We shall therefore arrive at length at a set glt ...,^B_1 of integral functions, in terms of
which all integral functions can be expressed so that the condition of dimensions is
satisfied.
It is this system which it was our aim to deduce.
* Chap. IV. § 39.
78] FUNDAMENTAL INTEGRAL FUNCTIONS. 109
Ex. For the surface associated with the equation yz = (x, l)2/> + 2 a^ integral functions
can in fact be represented in the form (x, l}^ + (x, l)AjVn where rjl=y->f-xm. If m>p + l
the dimension of ^ is m. In order to ascertain whether the condition of dimensions is
satisfied we enquire whether there exist any functions of the form x [a + a± (y + .vm)/xm],
wherein a, at are constants, which are finite for >r = oo, namely whether [a + a1(3/£m + !)]/£
can be an integral function of £.
Shew that this can only be the case when a + 0^ = 0. Putting kr = [-a-lt-al(y^m + l)]l^
it is clear that kr.rm~l = aly. Thus all integral functions can be represented in the form
(x, !).+(#, 1). y. Shew that the condition of dimensions is now satisfied.
78. There is one part of the process given here which has not been explained. Let
?;!, ..., r/n-! be integral functions, and let u denote a linear function of the form x — c. It
is required to find all possible functions of the form
wherein a, ..., an_l are constants, which are not infinite when w = 0. We suppose
ij!, ..., »?„_! to be such that the product of every two of them is expressible in the form
•v + vlrjl + ... + vn_lr)n_l, v, ..., yn_! being integral polynomials in x ; this condition is
always satisfied in the actual case under consideration.
The integral function //= a + a17?1-|-...+an _i»7»-i will satisfy an equation of the form
(H - HW] ...(#- fl») = Hn + A\Hn l + ...+ Kn _ ,H+ Kn = 0,
wherein A"i is an integral polynomial in a, ..., an_! of the ith order ; Ki is also an integral
polynomial in x. In order that H/u be an integral function it is sufficient that Kt be
divisible by u\ and when H/u is an integral function these n conditions will always be
satisfied. And it is easy to see that if Si denote the sum of the {th powers of the n values
of H which arise for any value of x, these conditions may be replaced by the conditions
that Si be divisible by u^ It is clear that it may not be an easy matter to obtain the
values of a, ..., an_!, which satisfy the conditions thus expressed.
But in fact these conditions can be reduced to a set of linear congruences, and event
ually to a set of linear equations for a, ..., an_i. We shall not give here the proof of this
reduction*, but give the resulting equations. For in many practical cases we can obtain
the results, geometrically or otherwise, in a much shorter way.
Let
/
denote in order of magnitude all the positive rational numerical fractions not greater than
unity, whose denominators are not greater than n ; each being in its lowest terms. Let
Tj!, ..., 77,. denote any linearly independent integral functions. Let 2 denote the sum of the
n values of a function which arise for any value of x. Determine all the possible sets of
values of the constants a, a1} ..., ar such that the congruence
2(a + a1771 + ... + ar77r)(c + c1771 + ... + cr77r) = 0 (mod. u)
is satisfied for all values of the quantities c, clt ..., cr. Substituting in the left hand the
value of x for which u = 0 and equating separately to zero the coefficients of c, clt ..., cr, we
obtain r-\-l linear equations for the constants a, «j, ..., ar. By these equations we can
* Which is given by Hensel, Acta Math. 18, pp. 284 — 292. His use of homogeneous variables
is explained below Chap. VI. § 85. But it is unessential to the theory of the reduction referred to.
110 ACTUAL ALGEBRAICAL DETERMINATION OF [78
express a certain number* of a, alf ..., ar in terms of the others ; denoting these others by
ft, ..., ft the function a + a^ + .-.-fa,.^ takes the form ftd + .-.+ftf*, wherein £lt ..., £,
are definite linear functions of 1, i^, ..., r)r with constant coefficients, and the equations in
question are then satisfied for all constant values of ft, ..., ft. We associate f the functions
CD •••) f« with the first term - of the series of fractions specified above. We proceed thence
7i
to deduce a set of integral functions associated with the next term of the series, — .
?i ~— 1
But in order to be able to describe the successive processes in as few words as possible, let
us assume we have obtained a set of integral functions £j, ..., £m which in the sense
employed are associated with\ the fraction e of the series, and wish to deduce a set of
functions associated with the next following fraction of the series, «'. Put down the con
gruence
2 (yi& + . ..+*»&») («i£i + .» +e,B&»)i-isO (mod. w^i).
Herein ylt ..., ym denote constants, {denotes in turn all positive integers not greater
than n which are exact multiples of the denominator of the fraction e, so that if is an
integer, \it' denotes the least integer which is not less than ie', and, for any proper value
of ij the congruence is to be satisfied for all values of the quantities ex, ..., em. It will be
found in practice that the left-hand side divides by u]if' :~1 for all values of y15 ...,ym,
%,..., em. If we carry out the division, then, in the result, substitute the value of x
which makes u=0, and equate separately to zero the coefficients of the ( . ) products of
\i— l/
e1, ..., em which enter on the left, we shall have this number of linear equations for
7u •••> ym- Solving these, and thereby expressing as many as possible of yx, ..., ym in
terms of the remaining, which we may denote by y/, ..., y'm>, yi£i + ... + ym£m will take a
form yi£i' + ...+y'm'gm', wherein y/, ..., y'm' are arbitrary constants, and £/, ..., gm> are
definite linear functions of £ls ..., £m. We say that £/, ..., %m> are associated with the
fraction e'.
This process is to be continued beginning with the case when e-- and ending with the
Yi
case when e' = l. The functions associated with the last term, 1, of the series of frac
tions, say G!, ..., Gk, are all the functions of the form a + alr)l + ... + an ^lrjn_l, wherein
a, als ..., an^l are constants, which are such that GJu, ..., Gk/u are finite when u=0.
For the case » = 3, of a surface of three sheets, the series is J, |, |, 1. The successive
congruences may therefore be denoted by
(S2) = 0 (mod. it), (S3) = 0 (mod. w2), (>S'2) = 0 (mod. w2), (S3) = 0 (mod. «3),
wherein (S^ denotes such an expression as 2 (yili + ...+y,»|m) (^i^i + '-'+^n^mY'1-
In fact 3 is the only integer not greater than 3 such that 3. ^ is integral and |3 . £| = 2.
And 2 is the only integer not greater than 3 such that 2 . £ is integral and 1 2 . § | =• 2 ;
finally 3 is the only integer such that 3 . § is integral, and 1 3 . 1 1 = 3.
For a surface of four sheets the fractions are
i, J, i, §, I, i.
* At most, and in general, equal to r.
\_
t In a certain sense the functions f1? ..., £, are all divisible by u«.
+ Divisible by xf, in a sense.
79] FUNDAMENTAL INTEGRAL FUNCTIONS.
We therefore have
111
1
<'
t such that it = integral
M
congruence
0
i
t = 2
i
(^sO (mod. %)
i
ft
?:=4
1^1 = 2
(*S4) = 0 (mod. «2)
1
i
t = 3
If =2
(£3) = 0 (mod. M2)
i
1
{=4
I!!-.
(52) = 0(mod. w2)
§
1
M
l!l = 3
(*S"3) = 0 (mod. u3)
1
1
;=4
141=4
(*S"4) = 0 (mod. w4)
It must be borne in mind that the results of the solution of each of the seven con
gruences of the sequence in the right-hand column, are here supposed to be substituted in
the next one : so that, for instance, the fourth congruence here may be quite other than a
slightly harder case of the first congruence.
Ex. Prove that for a surface of five sheets the congruences are, in order,
(I) (S2) = 0 (, «); (2) (S5) = 0 (, O; (3) (S4) = 0 (, «2) ; (4) (S3) = 0 (, «**) ; (5) (S5) = 0 (, w3);
(6)(S2) = 0(,«2); (7)(S4) = 0(,«3); (8)(^5) = 0(,^; (9) (S3) = 0 (,*»); (10) (S4) = 0 (, ««);
(II) (^ = 0 (,M6).
79. Ex. i. Prove for the equation y*=xz (x— 1) that A (1, y, y2, y3) = — 256 a6 (.r — I)3.
Shew that the equations
2 (a + a^y + azf + atf3)* = 0 (mod. (x - 1 )«),
where a, an a2, a3 are constants, and i is in turn equal to 1, 2, 3, 4, are only satisfied by
a = aj = o2 = a3 = 0.
Shew that the equations
2 O + fty + ft^ + flsy^O (mod. a*),
where 0, ft, ft, ft are constants, and i is in turn equal to 1, 2, 3, 4, require 0=ft = 0 and
leave ft and ft arbitrary. Hence y- , ^ are the only integral functions of the form
Shew that the equations
2 (-y+yi3/+72 ^+y3 7)* = *) (mod. **)
\ & x j
require y=y1 = y2=y3=o.
Prove that the dimensions of 1, y, y- , y- are 0, 1, 1, 2. Prove then that there is no
X X
function of the form
which is finite for x infinite.
112 EXAMPLES. [79
Hence 1, y, — , — are a fundamental system such as 1, gv, #2, 9z m Chap. IV. ; and
(C X
the deficiency of the surface is 1 + 1 + 2 •- (4 - 1 ) = 1 .
Ex. ii. In partial illustration of Hensel's method of reduction consider the case of the
equation
f - 3^/2 + 3y.v (x - 1 ) + x* (x - 1 )2 (O.^3 + 7.v2 + 5.r + 3) = 0,
for which the sums of the powers of y are given by
s = 3.r s = 3#2 + fix s = -
The determinant A (1, y, j/2) is divisible by .r3 and by (x — I)2, as appears on calculation.
By forming the equation satisfied by yl\x it appears that y^jx is an integral function.
Denote it by r). We consider now what functions exist of the form
(a + atf + a2 rf)l(x -1),
wherein a, a1? a2 are constants, which are integral functions.
The congruence (S2) = 2 (a + a^ + a^) (c + clty + e2^) = 0 (mod. .r-1) leads, considering
the coefficients of c, cn c2 separately, to the congruences
s1 + a2-0( ,.r- 1), a^
and therefore to the equations
which give a = 0, ax= — 3a2, and shew that the only function of the kind required is, save
for a constant multiplier,
(,-%)/(*- 1).
The other three congruences reduce then to conditions for this function ; for example,
the congruence (£3) = 0( , ^2) becomes
_x(x — 1) x— 1
But in fact, if we write g = (y^-Zxy}jx (x-\\ A = dx3 + 7.£2 + 5.v + 3, we immediately
find from the original equation that
g3 + §gi _ fy (A.x -$) + Azx(x-l) + 9 Ax = 0,
so that g is an integral function.
Apply the method to shew that y*jx is the only integral function of the form
Prove that the dimensions of the functions
are respectively 0, 3, 3.
Putting a? = 1/|, y/.r3 = A, examine whether there exists any integral function of £ of
the form
[a + aiA + 3a2 (A2-3£%)/£ (!
and deduce the fundamental integral functions.
The deficiency of the surface- is 3 + 3 -(3-1) = 4.
81]
CHAPTER VI.
GEOMETRICAL INVESTIGATIONS.
80. IT has already been pointed out (§ 9) that the algebraical equation,
associated with a Riemann surface, may be regarded as the equation of a
plane curve ; for the sake of distinctness we may call this curve the funda
mental curve. The most general form of a rational function on the Riemann
surface is a quotient of two expressions which are integral polynomials in
the variables (x, y) in terms of which the equation associated with the surface
is expressed. Either of these polynomials, equated to zero, may be regarded
as representing a curve intersecting the fundamental curve. Thus we may
expect that a comparison of the theory of rational functions on the Riemann
surface with the theory of the intersection of a fundamental curve with other
variable curves, will give greater clearness to both theories.
In the present chapter we shall make full use of the results obtainable
from Riemann's theory and seek to deduce the geometrical results as con
sequences of that theory.
81. The converse order of development, though of more elementary
character, requires much detailed preliminary investigation, if it is to be
quite complete, especially in regard to the theory of the multiple points
of curves. But the following account of this order of development may be
given here with advantage (§§ 81 — 83). Let the term of highest aggregate
degree in the equation of the fundamental curve f(y, #) = 0 be of degree n;
and, in the usual way, regard the equation as having its most general form
when it consists of all terms whose aggregate degree, in x and y, is not
greater than n; this general form contains therefore £(7i+l)(w + 2) terms.
Suppose, further, that the curve has no multiple points other than ordinary
double points and cusps, 8 being the number of double points and K of cusps.
Consider now another curve, ty (x, y) = 0, of order m, whose coefficients are
at our disposal. By proper choice of these coefficients in -v/r we can determine
xp- to pass through any given points of y, whose number is not greater than
the number of disposeable coefficients in -v|r. Let k be the number of the
prescribed points, and interpret the infinite intersections of /and >/r, in the
usual way, so that their total number of intersections is raw. Then there
B. 8
114 INTRODUCTORY SKETCH. [81
remain mn — k intersections of / and i/r which are determined by the others
already prescribed. We proceed to prove that if m > n — 3, and if we utilise
all the coefficients of -fy to prescribe as many of the intersections of -^ andy
as possible, and introduce further the condition that i/r shall pass once through
each cusp and double point off, then the number of remaining intersections
which are determined by the others will be p = ^ (n — 1) (n — 2) — B — K*, for
all values of m. For, if m ^ n, the intersections of ty with f are the same as
those of a curve
^ + Um_nf= 0,
wherein Um_n is any integral polynomial in the coordinates a; and y, in which
no term of higher aggregate dimension than m — n occurs. By suitable
choice of the ^ (m — n + 1) (m — n + 2) coefficients which occur in the general
form of Um-n we can reduce ^ (m—n + l)(m — n 4- 2) coefficients in \jr+ Um-nf
to zero-f". It will therefore contain, in its new form,
M + I = 1 + \m (m + 3) - £ (m - n + 1) (m - n + 2)
arbitrary coefficients. M is therefore the number of the intersections of ^r
with f which we can dispose of at will, by choosing the coefficients in >/r
suitably. Of these intersections, by hypothesis, 2 (8 + K) are to be taken
at the double points and cusps of the curve /. This can be effected by the
disposal of £ + K of the arbitrary coefficients. There remain then
1 + \m (m + 3) - \(m - n + 1) (m - n + 2) - 8 - tc
disposeable coefficients and mn — 2 (8 + K) intersections. Of these, therefore,
mn - 2 (8 + K) - [| m (in + 3) - \(m - n + 1) (m - n + 2) - B - K]
is the number of intersections determined by the others which are at our
disposal ; and this number is
£(n-l)(n-2)-(S + *).
In case m < n, of the mn — 2 (8 + K) intersections of ty with f, which are
not at the double points or cusps of y, we can, by means of the ^w(m+3)— 8— K
coefficients of ^r which remain arbitrary when ty is prescribed to vanish at
each double point and cusp, dispose of all except
mn - 2 (S + K) - [|ra (m + 3) - (8 + K)] ;
when m = n — 1 or n — 2 it is easily seen that this is the same as before.
82. Let us assume now that the polynomials which occur, as the nume
rator and denominator, in the expression of a rational function, have the
* Reasons are given, Forsyth, Theoi-y of Functions, p. 356, § 182, for the conclusion that this
number is the deficiency of the Riemann surface having / (y, x) = Q as an associated equation.
We shall assume this result.
t As, for instance, the coefficients of ym, ijm~\ ym^x, ..., yn, ynx, ..., ynxm'n, in which case
the highest power of y, in ^+ */„,_„/, that remains, is yn~l.
83] INTRODUCTORY SKETCH. 115
property here assigned to ty, of vanishing once at each double point and
cusp of/ Without attempting to justify this assumption completely, we
remark that if it is not verified at any particular double point, the rational
function will clearly take the same value at the double point by whichever of
the two branches of the curve / the double point be approached. As a
matter of fact this is not generally the case. Suppose then we wish to obtain
a general form of rational function which has Q given finite points of
/, A !,..., AQ, as poles of the first order. Draw through these poles,
Alt ..., AQ, any curve ty whatever, of degree greater than n — 3, which passes
once through each double point and cusp off. Then ty will intersect / in
mn - 2 (8 + *) - Q
other points Blt B2, .... Through these other points Blt _B2> ••• off, and
through the double points, draw another curve, S-, of the same degree as ty.
The curve ^ will in general not be entirely determined by the prescription
of the mn — 2 (8 + K) — Q points B1} -B2, ____ Let the number of its coefficients
which still remain arbitrary be denoted by q + 1. Then it would be possible
by the prescription of, in all,
mn - 2 (S + K) - Q + q
points of ^, to determine ^ completely. But by what has just been proved,
S- is determined completely when all but p of its intersections are prescribed.
Wherefore
mn - 2 (8 + K) - Q + q = mn - 2 (S + «) - p.
Hence Q — q = p, and ^ has the form
where X, \lt . . . , X^ are arbitrary constants and -v/r, S-1} . . . , ^9 are q + 1 linearly
independent curves, all passing through the mn — 2 (8 + K) — Q points
BI, B.2, ..., as well as through the double points and cusps; and the general
rational function with the Q prescribed poles will have the form
X + Xj .Rj + . . . + ^qRq ,
where Ri = ^/i|r ; and this function contains q + 1 arbitrary coefficients.
83. In this investigation, which is given only for purposes of illustration, we have
assumed that the prescription of a point of a curve determines one of its coefficients in
terms of the remaining coefficients, and that the prescription of this one point does not of
itself necessitate that the curve pass through other points ; and we have obtained not
the exact form of the Riemann-Roch Theorem (Chap. III. § 37), but the first approxima
tion to that theorem which is expressed by Q — q=p; this result is true for all cases only
when Q>n(n-3)-2(8 + ic).
We may illustrate the need of the hypothesis that the curves x//- and ^ pass through the
double points and cusps, by considering the more particular case when the fundamental
curve
/=(#» y)2 + (a» y\ + fa y)4 = °>
8—2
116 INTRODUCTORY. [83
wherein (x, y\ is an integral homogeneous polynomial in x and y of the second degree, etc.,
is a quartic with a double point at the origin #=0, y = 0. Since here «=4 and 8 + K = l,
we have
j»-$(»-l)(n-2)—&-*-i.3.2-l-«,
and therefore (in accordance with Chap. III. §§ 23, 24, etc.) there exists a rational function
having any three prescribed points as poles of the first order. Let us attempt to express
this function in the form ^/\/^, wherein ^, ^ are curves, of degree m, (m>l), which do not
vanish at the double point. Beside the three prescribed poles A^ A2, A3 of the function,
^ will intersect/ in 4m — 3 points Bv, JB2, .... The intersections with / of the general
curve ^ of degree TO, are the same as those of a curve
provided TO <•: 4, and are therefore determined by ^TO(m + 3)— £'(m — 4 + 1) (m — 4 + 2), or
4m — 3 of them. And it is easily seen that the same result follows when m=3 or 2.
Hence no curve ^ can be drawn through the points Slt B2, ... other than the curve \//-,
which already passes through them ; and the rational function cannot be determined in
the way desired. It will be found moreover that this is still true when the hypothesis,
here made, that ^ and ^- shall be of the same degree, is allowed to lapse. As in the
general case, this hypothesis is made in order that the function obtained may be finite for
infinite values of x and y.
A curve which passes through each double point and cusp of the fundamental curve /
is said to be adjoint. When / has singularities of more complicated kind there is a corre
sponding condition, of greater complexity. For example in the case of the curve
which, in the present point of view, we regard as a quartic, there is a singularity at the
infinite end of the axis of y. If, in the usual way, we introduce the variable z to make the
equation homogeneous, and then* put 3/ = l, whereby the equation becomes
we see that the branches are, approximately, given by z— ± &r2, namely there is a point of
self contact, the common tangent being 2 = 0. If we assume that it is legitimate to regard
this self contact as the limit of two coincident double points, we shall infer that the condi
tion of adjointness for a curve ^ is that it shall touch the two branches of / at the point.
For example this condition is satisfied by the parabola
which, by the same transformation as that above, reduces to
z = ax2 + bxz + cz2,
and it is obvious that the four intersections with / of this parabola, other than those at
the singular point, are determined by all but p of them, p being in this case equal to 1.
We shall see in this chapter that we can obtain these results in a somewhat different
way: the equation y2 = (l — x2) (1— £2#2) is a good example of those in which it is not
convenient to regard the equation as a particular case of a curve of degree equal to the
highest degree which occurs. Though this method, of regarding any given curve as a
particular case of one whose degree is the degree of the highest term which occurs in the
given equation of the curve, is always allowable, it is often cumbersome.
Ex. 1. Prove that the theorem, that the intersections with / of a variable curve ^ are
determined by all but p of them, may be extended to the case where / has multiple points
* This process is equivalent to projecting the axis y = 0 to infinity.
84] THE ASSOCIATED EQUATIONS. 117
of order k, with separated tangents, by assuming that the condition of adjointness is that
i//- should have a multiple point of order k — 1 at every such multiple point of /, whose
tangents are distinct from each other and from those of/. (In this case any such multiple
point of/ furnishes a contribution \k (k — 1) to the number 5-f K of/.)
Ex. 2. The curve y*=(x, 1)6 may be regarded as a sextic. Shew that the singular
point at infinity may be regarded as the limit of eight double points, and that a general
adjoint curve is
Ex. 3. Shew that for the curve tf = (x, l)2p + 2 a general adjoint curve is
For further information on this subject consult Salmon, Higher Plane Curves (Dublin,
1879). pp. 42—48, and the references given in this volume, § 9 note, § 93, § 97, § 112 note,
84. In the remaining analytical developments of this chapter we
suppose* the equation associated with the Riemann surface to be given in
the form
f(y, *) = y» + 3r-'(*, l)Al + ...+y(a;, 1)^ + 0, l)An=0,
so that y is an integral function of x. Let a- + 1 be the dimension of y ;
then cr + 1 is the least positive integer such that y/or+l is finite when x is
infinite; thus if we put #=l/f and y=i) !%*+*, a -f 1 is the least positive
integer, such that 77 is an integral function of £. This substitution gives
f(y, x)=Z-n{°+vF(r), a where
so that cr + 1 is the least positive integer which is not less than any of the
quantities
X1} X2/2, ..., Xn_j/(n - 1), \n/n.
Ex. 1. For the case
the dimension of y as an integral function of x is 3. Writing y = r)/£3, where •£=!/£, the
equation becomes
and Tj is an integral function of £ of dimension 2. In fact yi = v/!2=y/# satisfies the
equation
yf+yi*(x, i)3+yi(*, i)4+(-»» 1)5=°
and is finite when £ = oo , or # = 0.
Ex. 2. Shew that in the case in which the equation associated with the Riemann
surface contains y to a degree equal to the highest aggregate degree which occurs, o- = 0.
* Chap. IV. § 38.
118 HOMOGENEOUS VARIABLES. [84
Whenever we are considering the places of the surface for which x = oo ,
we shall consider the surface in association with the equation F(r), £) = 0 ;
and shall speak of the infinite places as given by f =0. The original equation
is practically unaffected by writing x — c for x, c being a constant. We may
therefore suppose the equation so written that at x = 0, the n sheets of
the surface are distinct ; and may speak of the places x = 0 as the places
85. By the simultaneous use of the equations f(y, #) = 0, F(r), £) =0,
we shall be better able to formulate our results in accordance with the view,
hitherto always adopted, whereby the places x = oo are regarded as exactly
like any finite places. But it should be noticed that both these equations
may be regarded as particular cases of another in which homogeneous variables,
of a particular kind*, /are used. For put x = ajfz, y = u/z<r+1; we obtain
) = g-n«r+D U(u; to, z\ where
U (u\ «, z) = un + M»-IS<H-I-*I (w, z)Kl + ...+ uz«l-v <<H-I)-AH_, (Wj ^^_i
and it is clear that U(u; &>, z) is changed into f(y, so) by writing u = y,
&> = #, z=l, and is changed into F(rj, £) by writing u = t), co = 1, z = g.
We may speak of w, z as forms, of degree 1, and suppose that they do not
become infinite, the values x = oo being replaced by the values z = Q. When
<», z are replaced by ta>, tz, t being any quantity whatever, u is replaced by
ta+1u, y and x remaining unaltered. We may therefore speak of u as a form
of degree a + 1.
Similarly U (u ; o>, z) is a form of degree n (o- + l), being multiplied by
£tt(<r+D when u, to, z are replaced by t*+1u, tta, tz respectively. That there
is some advantage in using such homogeneous forms to express the results of
our theory will sufficiently appear; but it seems proper that the results
should first be obtained independently, in order that the implications of the
notation may be made clear. We shall adopt this course.
Some examples of the change which our expressions will undergo when
the results are expressed by homogeneous forms, may be fitly given here : —
Instead off(y, x) we shall have U(u; w, z) which is equal to zn(<T+l]f(y, sc)\
instead of/' (y) we shall have U' (u) = z(n~1} (*+*>/' (y} ; instead of the integral
function *f ^, of dimension T{ + 1, an integral form ^ of degree r;+l, equal
to zTi+1gi, will arise ; since 2 (rf + 1) = n +p — 1, it is easy to see that the
determinant J A (1, glt..., <7n-i) is equal to ^2n+^~2 A (1, #1,..., g^). In
accordance with § 48, Chap. IV. the former determinant will have a factor
* This homogeneous equation is used by Hensel. See the references given in Chap. IV.
(§ 42). It may be regarded as a generalization of the familiar case when <r = Q.
t Chap. IV. § 42.
J Chap. IV. § 43.
86] HOMOGENEOUS VARIABLES. 119
(&)— cz)r corresponding to a finite branch place of order r where x = c, and a
factor z8 corresponding to a branch place of order s at x = oo . Further, if,
by the formula (H) of page 63, we calculate the form fa (u, co, z) from
glt ..., gn-i, as fa(x, y) is there calculated from glt ..., gn-\, it is easy to see
that we obtain a form, fa(u, w, z), which is equal to zin~1)(a+1)~(Ti+l} fa(x, y).
Hence also, if ult <olt zl denote special values of u, co, z, the integral
[zdw — o)dz fjr1 fa (u^o), z) + S/A^ fa (u, <*, z)gr(u1> o^, z,)
U1 (u) oaZi — a>iZ
wherein /j, = (bo) — az}l(bwi — az}), a and b being arbitrary constants, is equal to
dx . z^ <-+'> ^ fa (x, y) + 2/^ (z1/z)Tr+^fa(g;)
f
{ -j/\ & ? (if nr
I V J 4t&\ \W *Aj\
\t/ / •*• \
and is thus equal to
dx Ar1 fa (x,
where X = /jLzJz = (bx — a)/(bx1 - a).
If in this we put 6 = 0, we obtain the form which we have already shewn
to be part of the expression of an integral of the third kind (Chap. IV. p. 67).
But if we put 6=1, the integral is exactly what we have already deduced
(Chap. IV. p. 70, Ex. 1) by the ordinary process of putting x— !/(£- a)
and regarding % as the independent variable.
We may, if we please, further specialise the quantities co, z, of which
hitherto only the ratio has been used, supposing* them defined by
o)=x((x — c), z=I/(x—c), where c is a constant. Then &> — cz—\.
Ex. 1. The integral of the first kind obtained in Chap. IV. § 45, p. 67, can similarly
be written
Ex. 2. In the case y2=(x, l)2p + 2> wherein y is of dimension p + 1, the equation
U (u; u>, z) = 0 is
v? = (a>, 2)2P+2
obtained by putting y = u/zp'*'1, x=a>/z.
86. We shall be largely concerned here with rational polynomials which
are integral in x and y. The values of such a polynomial here considered
are only those which it has for values of y and x satisfying the fundamental
equation. We shall therefore suppose every integral polynomial in x and y
reduced, by means of the fundamental equation, to a form in which the
highest power of y which enters is yn~l, say to a form
* In this view u and z are functions. If we regard c as throughout undetermined, we may
regard these functions as having no definite infinities.
120 THE GRADE OF A POLYNOMIAL. [86
If herein we write y = 77/f°'+1, #= l/£, cr + 1 being, as before, the dimen
sion of y as an integral function of x, we shall obtain -v^- (y, x) = £~° ^ (77, £),
where "^ (77, £) is an integral polynomial in 77 and £ of which a representative
term is
^n+i £G-(n-i-i} (*+D-H(\t ^ i=Qt 1? ...... j (n_1}
and G is the positive integer equal to the greatest of the quantities
Thus (r is the highest dimension occurring for the terms of -fy (y, x),
and ¥ (77, £) is not identically divisible by £. The dimension of the integral
function -^ (y, #) may be G ; but if M* (77, £) vanish in every sheet at £ = 0,
the dimension of ty(y, x) will be less than G. For this reason we shall
speak of G as the grade of ty (y, x). It is clear that if all the values of TJ
for £ = 0 be distinct, that is, if Ff (77) do not vanish for any place £ = 0, the
polynomial M/" (77, £), of order ?i — 1 in 77, cannot vanish for all the n places
f= 0. In that case the grade and the dimension of ty (y, x} are necessarily
the same. Further, by the vanishing of one of the coefficients, a polynomial
of grade G may reduce to one of lower grade. In this sense a polynomial of
low grade may be regarded as a particular case of one of higher grade.
In what follows we shall consider all polynomials whose grade is lower
than (n — 1) a- + n — 3 or (n — 1) (cr + 1) — 2, as particular cases of polynomials
of grade (n — 1) cr + n — 3 : the general expression of the grade will therefore*
be (n — l)cr + n — 3 + r, or (n - 1) (<r + 1) + r — 2, where r is zero or a positive
integer. The most general form of a polynomial of grade (n — l)(cr+l) + r — 2
is easily seen to be
i/r (y, x) = y^ (x, l\_2 + yn~°- (x, l\_, + ...+ yn-^ (x, 1)^ + ...+(«, l)r-i
wherein the first line is to be entirely absent if r = 0, the first term of the
first line is to be absent if r = 1, and the first term of the second line is to be
absent if cr = 0.
Hence when r > 0, the general polynomial of grade (n — 1) cr -f n — 3 + r
contains
nr - 1 + £ (n - 1) (n - 2 + na)
terms, this being still true if cr = 0 ; but when r = 0, the general polynomial
of grade (n — 1) a + n — 3 contains
terms. This is not the number obtained by putting r = 0 in the number
obtained for r > 0.
* The number is written in the former way to point out the numbers for the common case
when (r = 0.
87] GENERALIZED ZEROS. 121
Further, putting
, x) = g-ct-w-c^-v-ripfa f),
and denoting the aggregate number of zeros of "SP (77, £) at £ = 0 by /z, it
is clear that the aggregate number of infinities of i/r (y, x) at x = oo is
[(n — 1) a- + n — 3 + r]n — p. Since ty (y, x) is only infinite for x = oo , this
is also the total number of zeros of ty (y, x). We shall find it extremely
convenient to introduce a certain artificiality of expression, and to speak
of the sum of the number of zeros of ty(y, a) and the number of zeros of
^ C7?. |) at £ = 0 as the number of generalized zeros of -\/r (y, x). This number
is then n (n - 1) (a + 1) -f n (r - 2).
If by a change in the values of the coefficients in ty (y, x), "V (77, £ )
should take the form If^i (77, £) where ^1(77, &)qJ£c(,f7 integral polynomial
in 77 and £ so that ^ (y, x) is w-> . • { l^/,/v ^ (77, £), the sum of
,, -i f n •/ „ >~iial for the surface \y> %, ^
the number of finite zeros of x. „ fc-"p. of zeros of YI (77, f )
« '--'.9\ ..9_ 2 I Vi^fZ- ? »\»» >/
is »i(w- l)(o-+l) + n(r-3). i^ut, a.^ , c,; is equal to ^(77, f),
the number of zeros of M/" (77, ^) at ^ = 0 is ?i more than the number of zeros
of ^ (77, f ) at £ = 0. Hence the sum of the number of finite zeros of ty (y, x)
and the number of zeros of M/" (77, ^) at £ = 0, is still equal to
n (n - 1) (o- + 1) + ?i (r - 2).
^r. i. The number n (n- 1) (<r + l) + ?t (r-2) is clearly the number of zeros of the
integral form
Ex. ii. The generalized number of zeros of/' (?/), for which r = 2, is 71 (?i — 1) (o- + 1).
Ex. iii. The general polynomial of grade d, < (n — 1) <r + ?i — 3, contains
terms'
^(x-) being the greatest integer in *\ Its generalized number of zeros is nd.
87. We introduce now a certain speciality in the integral polynomials
under consideration, that known as adjointness.
An integral polynomial •x/r (y, x) is said to be adjoint at a finite place
(x = a, y = b) when the integral
fyu/.a-1
J f(y}
is finite at this place. If t be the infinitesimal at the place (Chap. I. §§ 2, 3)
the condition is equivalent to postulating that the expression
f'(y)
122 ADJOINT POLYNOMIALS. [87
shall be finite at the place; or again equivalent to postulating that the
expression
(x-a)^ (y, x)
f(y)
shall be zero at the place, to the first order at least.
As a limitation for the polynomial i/r (y, x\ the condition is therefore
ineffective at all places where /' (y) is not zero. And if at a finite place
where /'(y) vanishes, i + w denote the order of zero of /' (y\ w + I being
the number of sheets that wind at this place*, the condition is that ty (y, x}
vanish to at least order i at the place. We shall call ^i the index of the
place ; the condition of adjointness is therefore ineffective at all places
of zero index.
If i/r (y} x} be of grade (n — 1) a- + n — 3 + r, and
t (V, *) = {-(^'-(K-V-r V (rj, fr
the condition of adjointness of i/r (y, x) for infinite places, is that, at all
places | = 0 where F' (ij) = 0, the function
should be zero, to the first order at least. It is easily seen that this is
the same as the condition that the integral
fl
J of
dx
should be finite at the place considered.
When the condition of adjointness is satisfied at all finite and infinite
places where f (y) = 0 or F' (77) = 0, the polynomial i/r (y, x) is said to be
adjoint. If II (x — a) denote the integral polynomial which contains a
simple factor corresponding to every finite value of x for which f'(y) vanishes,
and if N denote the number of these factors, it is immediately seen that the
polynomial ty (y, x) is adjoint provided the function
is zero, to the first order at least, at all the places where /' (y} = 0 or
F' (r,) =0.
Ex. i. For the surface associated with the equation
/ (y, #) = to y\ + to y )3 + to y\ =-• °
there are two places at x=Q, at each of which y = 0. At each of these places /'(y) vanishes
to the first order, and w=0. Hence the condition of adjointness is that ^(y, x) vanishes
It is easy to see that i is not a negative integer. Cf. Forsyth, Tlieory of Functions, p. 169.
88] GENERAL FORM OF PLUCKER'.S EQUATIONS. 123
to the first order at each of these places. The general adjoint polynomial will therefore
not contain any term independent of x and y.
Ex. ii. For the surface
y* _ f [( 1 + #5) & + !] + tfyA = 0
there are two places at #=0, at each of which y is zero of the second order : they are not
branch places. At each of these /'(y) vanishes to the second order.
The dimension of y is 1, and the general polynomial of grade (n - 1) <r + n - 3 + 1 or 2, is*
Af + By + C+ x [Dy + Ex + F].
In order that this may vanish to the second order at the places in question, it is sufficient
that (7=0 and F=0. Then the polynomial takes the form
By + Ay2 + Dxy + Ex\
and if we put x/rj for x and !/»/ for y this becomes, save for a factor 17 ~2,
which is therefore an adjoint polynomial for the surface
Compare § 83.
Ex. iii. Prove that the general adjoint polynomial for the surface
y2 = (#- a)3,
is y(x, l)r_2 + (^-a) (x, l)r_1 = 0.
(The index of the place at x= a is 1.)
88. Since the number of generalized zeros of f'(y} is n(n— l)(o- + 1),
(§ 86, Ex. ii), we have, in the notation here adopted,
2(i+w) = n(n-I)(<r + l),
or if / denote 2t and W denote 2w, the summation extending to all finite
and infinite places of the surface
/+ W = n(n-l)(<r+l).
Hence, as^
W = 2rc + 2j» - 2,
we can infer
p = Kn-l)(n-2 + "<r)-J/,
shewing that / is an even integer.
Further if X denote the number of zeros of an adjoint polynomial
•fy (y, x), of grade (n — 1) <r + n — 3 + r, exclusive of those occurring at places
where /' (y) = 0 or F' (rj) = 0, and calculated on the hypothesis that the
adjoint polynomial vanishes, at a place where f'(y) or F' (77) vanishes, to an
order equal to twice the index of the place J, we have the equation
X + / = n (n - 1) (a- + 1) + n (r - 2).
* § 86 preceding.
t Forsyth, Tiieory of Function*, p. 349.
£ So that a place of index ^i where ^ (y, .r), or ^ (77, £), vanishes to order i + \, will furnish a
contribution X to the number X.
124 TRANSFORMATION TO CASE OF SIMPLE NODES. [88
Thus, as
/ = n (n - 1) (a- + 1) - 2 (n - 1) - 2p,
we have
X = nr + 2p - 2 ;
and this is true when r = 0.
These important results may be regarded as a generalization of some of
Pliicker's equations* for the case cr= 0.
Ex. i. The number of terms in the general polynomial of grade (n-l) o- + n-3 + r was
proved to be \(n - 1) (n - 2 + w<r) + nr - 1 or |(w - 1) (n - 2 + wo-), according as r > 0 or r = 0.
This number may therefore be expressed as p + \I-\- nr-l or p + \I in these two cases.
Ex. ii. It is easy to see, in the notation explained in § 85, that the homogeneous form
A (1, u, w2, ... , un~l) is of degree n (n - 1) (0-+ 1) in &> and z, and the form A (1, g^ ... , ffn,j)
of degree W. The quotient A(l, u, ..., un-l)/&(l, glt ... , gn_^ is (§ 43) an integral form
in o>, z, which, by an equation proved here, is of degree 7. It is the square of an integral
homogeneous form v whose degree in u>, z together is |/.
Ex. iii. It can be proved (compare § 43 b, Exx. 1, 2, and § 48 ; also Harkness and
Morley, Theory of Functions, pp. 269, 270, 272, or Kronecker's original paper, Crelle, t. 91)
that if for y we take the function
wherein X, Xt, ... , Xn_1 are integral polynomials in x, of sufficient (but finite) order, the
polynomial v occurring in the equation,
A(l, y, ... , y»-1) = V2A (1, fflt ... , <?„_!),
cannot, for general values of the coefficients in X, X:, ... , Xn_:, have any repeated factor, or
have any factor which is also a factor of A(l, g^ ..., ffn_1). And the inference can be
madef that for this dependent variable y, there is no place at which the index is greater
than ^, and no value of x for which two places occur at which f'(y\ or F'(T}), is zero.
89. We proceed, now, to shew the utility of the notion of adjoint
polynomials for the solution of the problem of finding the expression of
a rational function of given poles.
Let R be any rational function, and suppose, first, that none of the finite
poles of R are at places where f'(y) = 0. Let ^r be any integral polynomial,
chosen so as to be zero at every finite pole of R, to an order at least as high
as the order of the pole of R, and to be adjoint at every finite place where
/' (y) vanishes. Denote the integral polynomial II (x — a), which contains a
linear factor corresponding to every finite value of x for which /' (y) vanishes,
by fj,. Then the rational function
* Salmon, Higher Plane Curves (Dublin, 1879), p. 65.
t See also Noether, Math. Annal. t. xxiii. p. 311 (Rationale Ausfiihrung, u. s. w.), and Halphen,
Comptes Rendus, t. 80 (1875), where a proof is given that every algebraic plane curve may be
regarded as the projection of a space curve having only one multiple point at which all the
taugents are distinct. But see Valentiner, Acta Math., ii. p. 137.
89] EXPRESSION OF RATIONAL FUNCTION OF GIVEN POLES. 125
is finite at all finite places where R is infinite, and is finite, being zero,
at every finite place at which /' (y) = Q. If 2/1, ... , yn denote the n values
of y which belong to any value of x, and c be an arbitrary constant, the
function
» (c-y,)(c-y,)...(c-yn) ^ ^ ^
i = \ c ~ yi
is a symmetrical function of ylt ..., yn and, therefore, expressible as a rational
function in x only; moreover the function is finite for all finite values of
x and, therefore, expressible as an integral polynomial in x. Since this
polynomial vanishes for every finite value of x which reduces the product
p to zero, it must divide by JJL. Finally, the function is an integral polynomial
in c, of degree n — l. Hence we have an equation of the form
wherein A0, Alt ..., -4n-i are integral polynomials in x.
Therefore, putting c = ?/;, recalling the form of the function A (y, x), and
replacing y{ by y, we have the result
which we may write in the form
R =
^ being an integral polynomial in x and y.
Since
(x — a) ^ _ p (x — a) ty
~7rW ~7W
^s, like ty, is adjoint at every finite place where y" (y) vanishes.
Suppose, next, that the function R has finite poles at places where f (y)
vanishes. Then the polynomial ty is to be chosen so that R (x — a) ty/f (y)
is zero at such a place, a being the value of x at the place. This may be
stated by saying that -^ is adjoint at such a place and, besides, satisfies
the condition of being zero at the place to as high order as R is infinite.
Corollary. Suppose R to be an integral function ; and for a finite place,
x = a, y=b, where /' (y) vanishes, suppose t + 1 to be the least positive
integer such that (x - a)t+l/f' (y) has limit zero at the place. Then the
polynomial -^ of the preceding investigation may be replaced by the product
II (x - a)', extended to all the finite values of x for which /' (y) is zero.
Hence, any integral function is expressible in the form
126 RATIONAL FUNCTION OF GIVEN POLES. [89
where S- is an integral polynomial in x and y, which is adjoint at every finite
place where f (y) vanishes.
If the order of a zero of f (y) be represented as before by i + w, it is
clear that the corresponding value of t + 1 is the least positive integer for
which (t + I)(w + l)>i + w, or, for which t>(i — l)/(w+l). Hence the
denominator IT (x — a)* only contains factors corresponding to places at which
the index | i is greater than zero ; if the index be zero at all the finite places
at which f (y) vanishes, every integral function is expressible integrally.
It does not follow that when the index is zero at all finite places, the functions
I, y, ... , yn~\ form a fundamental system of integral functions for which the condition of
dimensions is satisfied. For the sum of the dimensions of 1, yt ... , yn~l is greater than
p+n—l by the sum of the indices at all the places # = oo .
It is clear that if R be any rational function whatever, it is possible
to find an integral polynomial in x only, say X, such that \R is an integral
function. To this integral function we may apply the present Corollary.
The reader who recalls Chapter IV. will compare the results there obtained.
90. Let the polynomial ty be of grade (n—l)<r+n—'3+r, and the
polynomial ^ of grade (n — 1) a + n — 3 + s, so that
^ _ fc- (n-i) o— (n-3) -r \p- ^ _ fc- (n-i) o— (n-3) -s (S)
and JB = f-«0/¥,
®, "^ being integral polynomials in 77 and £.
If R have poles for £ = 0, it will generally be convenient to choose the
polynomial ^ so that R^f is finite at all places £ = 0 ; if F' (77) vanish for
any places £= 0, it is also convenient, as a rule, to choose -^ so that %ty/F'(r))
vanishes at every place f = 0 where F' (77) vanishes, namely, so that i/r
is adjoint at infinity. When both R is infinite and F' (77) vanishes at a
place where f = 0, we may suppose ty so chosen that gRW/F' (77) is zero at
the place. Let ty be chosen to satisfy these conditions. Then, since
RW, = R-^r . ^(n~l)<7+n~3+r, is finite at every place, except f=oo, and
(1 — a£) W/F' (77), = |:r~1 (x — a) ^rjf ' (y), vanishes at every place x = a, y = b,
where # is finite, at which f (y) vanishes, except £ = oo , it follows, as here,
that R can be written in a form
R=®,/V>
wherein ©x is an integral polynomial in 77 and £.
Hence ®j = %r~s®, and therefore r — s is not negative : namely, the
polynomial ^ which occurs in the expression of a rational function in the
form R = ^/"fr, is not of higher grade than the denominator i/r, provided
-\Jr be chosen to be adjoint at infinity, and, at the same time, to compensate
the poles of R which occur for x = oo . Since a polynomial of low grade
91] ^-POLYNOMIALS. 127
is a particular case of one of higher grade we may regard ^ and -fy as of the
same grade.
Hence we can formulate a rule for the expression of a rational function of
assigned poles as follows — Choose any integral polynomial i/r which is adjoint
at all finite places and is adjoint at infinity, which, moreover, vanishes at
every finite place and at every infinite place* where R is infinite, to as high
order as that of the infinity of R. If a pole of R fall at a place where
f'(y), or F' (77), vanishes, these two conditions may be replaced by a single one
in accordance with the indications of the text. Then, choose an integral
polynomial S-, of the same grade as i/r, also adjoint at all finite and infinite
places, which, moreover, vanishes at every zero of the polynomial -fy other than
the poles of R, to as high order as the zero of ty at that place. Then the
function can be expressed in the form
91. We may apply the rule just given to determine the form of the
integrals of the first kind.
If v be any integral of the first kind, dvjdx is a rational function having
no poles, for finite values of x, except at the branch places of the surface. If
a be the value of x at one of these branch places, the product (x — a) dvjdx
vanishes at the place. Hence we may apply to dvjdx the same reasoning
as was applied to the function A (y, x) in § 89, and obtain the result, that
dv/dx can be expressed in the form
dv yn~*AQ + yn~2A1 + ... + yA^ + An_,
dx f(y)
wherein A0, ..., An^ are integral polynomials in x. Denote the numerator
by <f>, and let its grade be denoted by (n - 1) cr + n — 3 + r ; then
But, as a function of £, dv/dg has exactly the same character as has dv/dx
as a function of x. Thus by a repetition of the argument F' (77) dv[d% is
expressible as an integral function of rj and £. Thus r is either zero or
negative.
(IV
Wherefore, /' (y} -5- is an integral polynomial in x and y, of grade
(n — I)<r + n — 3 or less. It is clearly adjoint at all finite places, and,
reckoned as a particular case of a polynomial of grade (n — 1) <r + n— 3, it is
clearly also adjoint at infinity.
Conversely, it is immediately seen, that if $ be any integral polynomial of
* That is, if the polynomial be i//, of grade («- 1) <r + n-3 + r and ^ = ^|-(»-i)»1-(n-S)-r) >j>
vanishes at £ = 0 to the order stated. A similar abbreviated phraseology is constantly employed.
128 NUMBER OF UN ASSIGNED COEFFICIENTS IN ADJOINT POLYNOMIAL. [91
grade (n — 1) «r + n — 3, which is adjoint at all finite and infinite places, the
integral
/•/ xy>
I *i , * tt-U/j
is an integral of the first kind.
Corollary. We have seen that the general adjoint polynomial of grade
(n— l)o- + w — 3 contains p + ^I terms, and we know that there are just
p linearly independent integrals of the first kind. We can therefore make
the inference
The condition of adjointness, for a polynomial of grade (n— 1) <r + n — 3,
is equivalent to ^1 linearly independent conditions for the coefficients of the
polynomial, and reduces the number of terms in the polynomial to p.
92. We have shewn that a general polynomial of grade (n — l)o-+n— 3 + r
is of the form
^n-3+r = y»-i (x, l)r_2 + y»-s (x, l)r_a + ...+y(x, !),_! + (x, l)r_x
We shall assume in the rest of this chapter that the condition of adjoint-
ness for a general polynomial of grade (n — l)cr + n — 3 +r is equivalent
to as many independent linear conditions as for a general polynomial of
grade (n — 1) a + n— 3. Thence, the general adjoint polynomial of grade
(n— l)cr + n-3 + r contains nr — 1 + p terms.
Further we shewed that the adjoint polynomial of grade (n — 1) a- + n — 3
has %p — 2 zeros exclusive of those falling at places where f (y) = 0, or
F'(r,) = 0.
Hence, the 2p — 2 zeros of the differential dv (Chap. II. § 21) are the
zeros of the polynomial f (y) dv/dx, exclusive of those where f (y) = 0, or
*"(,) = <>.
It is in fact an obvious corollary from the condition of adjointness that
dV/dt = Wf(y)]tj
dx
only vanishes when 0 vanishes. For, at a place where /' (y)=0, (/> vanishes i times, -j-
vanishes w times, and/'(y) vanishes i+w times.
Ex. i. For the surface associated with the equation
where (x, 1)1? ... are integral polynomials in x of the degrees indicated by their suffixes,
<r = 0; and the general polynomial of grade (n—l)<r + n — 3 or 1, is of the form (§ 86)
Ay+Bt+C.
The indices of the places where /'(y) = 0 are easily seen to be everywhere zero — there
are no places, beside branch places, at which f'(y) vanishes. Hence p is equal to the
number of terms in this polynomial, or p = 3. And this polynomial vanishes in 2/>-2 = 4
places. These results may be modified when the coefficients in the equation have special
values.
92] DEFICIENCY OF WEIERSTRASS'.S CANONICAL SURFACE. 129
Ex. ii. For the more particular case when the equation is
(*, 1)2=0
there are two places at x=Q at which y=0. For general values of the coefficients in the
equation these are not branch places and f'(y) vanishes to the first order at each ; the
index at each place is therefore \i where i=l, and the condition for adjointness of the
general polynomial of grade 1, is that it shall vanish once at each of these places. These
conditions are equivalent to one condition only, that (7=0. Hence, as there are no other
places where the index is greater than zero, the general integral of the first kind is
and p = 2; the polynomial Ay + Rv vanishes in 2p- 2 or 2 places other than the places
#=0, y=Q at which /'(y)=0.
Ex. iii. In general when the equation of the surface represents a plane curve with a
double point, the condition of adjointness at the places which correspond to this double
point, is the one condition that the adjoint polynomial vanish at the double point*.
Ex. iv. Prove that for each of the surfaces
l)2 + (^ 1)4=0,
there is only one place at infinity and the index there, in both cases, is 1.
Shew that the index at the infinite place of Weierstrass's canonical surface f is in all
cases
-r-l),
where
means the least integer greater than r/a, and that the deficiency is given by
where /' denotes the sum of the indices at all finite places of the surface.
Of. Camb. Phil. Trans, xv. iv. p. 430. The practical method of obtaining adjoint poly
nomials of grade (n— 1) <r+n — 3 which is explained in that paper (pp. 414 — 416) is often of
great use.
Ex. v. In the notation of Chap. IV. the polynomial
(X, I)*'"1 $!+...+(#, Ifn-l-1 <K-i
is an adjoint polynomial of grade (n - 1) v + n — 3.
Ex. vi. We can prove in exactly the same way as in the text that an integral of the
third kind infinite only at the ordinary finite places (xlt yj, (#/, y^\ at the former like
C\og(x-xl) and at the latter like - C log (x - #/), C being a constant, can be written in
the form
where ^ is an adjoint integral polynomial in x and y, of grade (n-l)<r + n—l, which
* The sum of the indices at the k places of the surface corresponding to an ordinary fc-ple
point of the curve is p (k - 1) ; the index at each of the places is in fact %(k - 1). Cf. § 83, Ex. i.
t Chap. V. § 64.
B. 9
UNIVERSITY
130 INTEGRAL OF THIRD KIND. [92
vanishes at the (n — 1) places x=xl where ?/ is not equal to yl and at the (ji-1) places
x=x± where y is not equal to yx'. Putting \^ in the form
+ (x- x^ (x - x^ (RQy» - 1 + Rtf - 2 + . . . + Rn _ J,
where C0, ..., C'n_1, C^', ..., C'n_l are constants, it follows, since (x-x^}yn~l is of grade
(n — l)<r+n, arid (^0yn~1 + Jff1yn~2 + ... + ^n_1) (x-x^) (x-x^ is of grade (n-l)<r + n+l
at least, that /£0 is zero and C0 — 0Q'. Further, if the equation associated with the surface
be written
and Xi (x) denote
it follows, from the condition for ^ which ensures that the integral P is not infinite at
all the n places x=xly that the factors of the polynomial
are the same as those of/(y, x}j(y — y^ or of
Hence, save for a constant multiplier, P has the form
/dx
^[fo^-teJtf-ttf-v'te IXr-i+y"-3^, l)*r+... + (#, l)(B_l)<r + n-3],
where (x, #x) denotes
[yn-1+yn-2XlK) + -+Xn-l(^l)]/(^-^l),
so that (x, #!) = (#!, A'), and (x, #/) denotes a similar expression.
A general polynomial ^ of grade (n-l)tr + n-l contains 2n-l more terms than a
general polynomial of grade (n- 1) a- + n — 3. In accordance with the assumption made in
§ 92 the general adjoint polynomial ^ of grade (« — 1) <r+n— 1 will contain 2n— l+p
terms. The condition that >//• vanishes in the 2»-2 places x=xly x=x^ other than those
where y=yl, y=y\ respectively, will reduce the number of terms to p + 1. This is exactly
the proper number of terms for a general integral of the third kind (cf. § 45, p. 67). The
assumption of § 92 is therefore verified in this instance.
The practical determination of an integral of the third kind here sketched is often very
useful. In the hyperelliptic case it gives the integral immediately.
Ex. vii. Prove that if the matrix of substitution Q occurring on p. 62, in *he equation
(1, y, y\ ... , y»-i) = a (!,&..., #„-,),
— 1
be denoted by Qx, and the general element of the product-matrix ^Q^ be denoted by
cr,g, and if, for distinctness of expression, we denote the elements
x»-i (*)• x*-* (*), - • xi (#)» !> !> yi, yi*> - . ^i"'1!
respectively by
UD UZ) ••' > Mn-l> ^n> *i> *"2> 3' ••• » n'
then the function
</>0 (a?) + 0! (ar) ^ (»i)-f- ... +4i»-i( (*)F»-1 (*i)i
which occurs in the expression of an integral of the third kind given in § 45, is equal to
cnulkl + ...+ciiuiki + ... + crturk,+ctrugkr+....
This takes the form <ul&1 + ...+unlkn obtained in Ex. vi. when crs=0 and crt=l, namely
when Q is a constant. This condition will be satisfied when the index is zero at all finite
and infinite places.
92] INTEGRAL OF THIRD KIND. 131
Ex. viii. Prove for the surface associated with the equation
y*+y*fa \\+y(x, l), + fa 1)4 = 0,
that the condition of adjointness for any polynomial is that it vanish to the second order
at the place £ = 0.
Thence shew that the polynomial
(x - #/) |>2 +yXl fa) + Xz fa)] - (x - xj [y* +y\i fa) + X2 «)]
+ ( Ay + Bx* + Cx+D)(x-xl)(x- #/)
is adjoint provided B=Q ; and thence that the integral of the third kind is
+yXl fa)+Xz fa) y* +yxifa')+x^fa) . . , / , 7)1
Ex. ix. There is a very important generalization* of the method of Ex. vi. for forming
an integral of the third kind. Let /z be any positive integer. Let a general non-adjoint
polynomial of grade /z be chosen so as to vanish in the two infinities of the integral, which
we suppose, first of all, to be ordinary finite places. Denote this polynomial by L. It
will vanish f in ?i/z-2 other places Blt J52, .... Take an adjoint polynomial \^, of grade
(n — l)o-+n — 3-{-/z, chosen so as to vanish in the places Blt B2, .... The polynomial will
presumably contain (§ 92) n^— 1 +p— (tip — 2) or p + l homogeneously entering arbitrary
coefficients, and will vanish (§ 88) in np + Sp — 2 — (np — 2) or 2p places other than the
places Blt Z?2, ... and places where f'(y), or F' (9), vanishes. Then the integral
is a constant multiple of an elementary integral of the third kind.
The proof is to be carried out exactly on the lines of the proof of the form of an
integral of the first kind in § 91, with reference to the investigation in § 89.
Further as we know (§ 16) that dPfdx is of the form
C (dP\dx\ + Xx (dv^dx) -f . . . + Xp (dvp/dx),
where C, Xu ... , Xp are arbitrary constants, (dPjdx\ is a special form of dPjdx with the
proper behaviour at the infinities, and vlt ... , vp are integrals of the first kind, it follows
that the polynomial \^, which is an adjoint polynomial of grade (n — l)<r-fn — 3 + /x, pre
scribed to vanish at all but two of the zeros of a non-adjoint polynomial L of grade /z, is of
the form
where \^0 is a particular form of >//• satisfying the conditions, and 0 is any adjoint poly
nomial of grade (n— 1) <r+n — 3 ; for this is the only form of -^ which will reduce dPjdx to
the form specified.
Ex. x. Shew that if in Ex. ix. one or both of the infinities of the integral be places
where f'(y) = 0, the condition for L is that it vanish to the first order in each place.
Ex. xi. For the case of the surface associated with the equation
* Given, for <r = 0, /* = !, in Clebsch and Gordan, Abel. Functionen (Leipzig, 1866), p. 22, and
Noether, "Abel. Differentialausdriicke," Math. Annal. t. 37, p. 432.
t Counting zeros which occur for x = oo , or supposing all the zeros to be at finite places.
Zeros which occur at x = oo are to be obtained by considering £^L, which is an integral polynomial
in f and 17 (§ 86).
9—2
132 ANY RATIONAL FUNCTION BY INTEGRAL FUNCTIONS. [92
for which the dimension of y is 1, let us form the integral of the third kind with its
infinities at the two places # = 0, y—Q by the rules of Exs. ix. and x. ; taking /x = l, the
general polynomial of grade 1 which vanishes at the two places in question is \x + py.
The general polynomial of grade n - 3 + /i, or 2, is of the form ax* +by2 + 2 to/ + 2gx + 2/y + c.
In order that this may be adjoint, c must vanish ; in order that it may vanish at the two
points, other than (0, 0) at which \x + ny vanishes, it must reduce to the form
Hence the integral of the third kind is I (Ax + By + C) dx/f (y). (Of. § 6 |8, p. 1 9.)
Ex. xii. Obtain the other result of § 6 /3, p. 19 in a similar way.
Ex. xiii. It will be instructive to compare the method of expressing rational functions
which is explained here, with a method founded on the use of the integral functions
obtained in Chap. IV. We consider, as example, the case of a rational function which has
simple poles at kl places where x=al, k.2 places where x=a.2, ..., Jcr places at x = ar, and for
simplicity we suppose all these values of x to be finite, and assume that the sheets of the
surface are all distinct for each of these values of x. If R be the rational function, the
function (x-a^)...(x -</,.) R is an integral function of dimension r, and is expressible in
the form
this form contains (r+l) + (»--T1) + ... + (?i-Tn_1) ornr-p + l coefficients ; these co
efficients are not arbitrary, for the function (x-al)...(x — ar) R must vanish at each of the
n — i\ places x—a-^ where R is not infinite, and must vanish at each of the places x=a2
where R is not infinite, and so on. The number of linear conditions thus imposed is
m-(kl+Li + ...+kr) or rn-Q, if Q be the total number of poles of the function R.
Hence the number of coefficients left arbitrary is nr— p + l-(nr- Q) or Q-p + l ; this is
in accordance with results already obtained.
Ex. xiv. If the differential coefficients of r + 1 linearly independent integrals of the
first kind vanish in the Q poles, in Ex. xiii., the conditions for the coefficients are equi
valent to only nr— Q- (T+ 1) independent conditions.
93. Let Al, ... , AQ be Q arbitrary places of the Riemann surface. We
shall suppose these places so situated that a rational function exists of which
they are the poles, each being of the first order*. This is a condition which
is always satisfied f when Q >p. The general rational function in question is
of the form
X + \iZ1 + . . . + \qZq,
wherein \, \ , . . . , \q are arbitrary constants and Zl , . . . , Zq are definite
rational functions whose poles, together, are the places Al, ..., AQ.
The number q is connected with Q by an equation
Q-q = p-T-i,
where T + 1 is| the number of linearly independent linear aggregates of the
form
* We speak as if the poles were distinct. This is unimportant.
+ Cf. Chap. III. % Chap. III. §§ 27, 37.
93J STRENGTH OF A SET OF PLACES. 133
which vanish in Al} ..., AQ. This aggregate is the differential coefficient, in
regard to the infinitesimal at the place x, of the general integral of the
first kind. We have seen* that this differential coefficient only vanishes
at a zero of the integral polynomial of grade (n— 1) <r + n — 3, which occurs
in the expression of the integral of the first kind. Hence T + 1 is the
number of linearly independent adjoint polynomials of grade (n — 1) <r + n — 3
which vanish in the places A1} ... , AQ ; in other words, r + 1 is the number of
coefficients in the general adjoint polynomial of grade (n— l)cr + n-3
which are left arbitrary after the prescription that the polynomial shall
vanish in Al, ... , AQ.
Now we have proved that if any adjoint polynomial ty, of grade
(n — l)<r + n — 3 + r be taken to vanish at the places A1} . .., AQ^, its other
zeros being Bl , . . . , BR, where J R = nr + 2p - 2 - Q, and ^ be a proper general
adjoint polynomial of grade (n — l)cr + n — 3 + r vanishing at 51(..., BR,
any rational function having Alt ..., AQ as poles, is of the form ^r/^f. Hence
the rational functions Z1} ..., Zq are of the forms ^a/^, ..., \/^, and the
general form of an adjoint polynomial of grade (n — I) <r + n — 3 + r vanishing
at BH ..., BB must be
wherein X, X1} ..., X5 are arbitrary constants, and ty, ^i,..., % are special
adjoint polynomials of grade (n— 1) <r + w — 3 + r which vanish in B1} ... , BR,
some of them possibly vanishing also in some of Alt ..., AQ.
Since the general adjoint polynomial ^ of grade (n — 1) o- + n — 3 +r
contains nr—l+p arbitrary coefficients, and these, in this case, by the
prescription of the zeros Blt ..., BR for S-, reduce to q + 1, we may say that
the places Blt ..., BK, as determinators of adjoint polynomials of grade
(n— l)a+n— 3+r, have the strength nr— l+p— q— l,or JR— (p — 1) + Q — q — 1,
or R — (r + 1). And, calling these places B1} ..., BR the residual of the
places Al} ..., AQ, because they are the remaining zeros of the adjoint
polynomial ^ of grade (n — 1) cr + n — 3 + r which vanishes in Alt..., AQ,
we have the result : —
When Q places Al ..... AQ have ilie strength p — (r + 1) or Q — q as
determinators of adjoint polynomials of grade (n— 1) <r + n — 3, their residual
of R = nr + 2p — 2 - Q places, which are the other zeros of any adjoint
polynomial of grade (n — l)o- + ?t — 3 + r prescribed to vanish in the places
Alt...t AQ, have the strength R — (r+l) as determinators of adjoint poly
nomials of grade (n — 1) <r + n — 3 + r.
Particular cases are, (i), when no adjoint polynomial of grade (n-l)o- + n — 3 vanishes
in Alt ..., Aa; then the places B1,...,Bll have a strength equal to their number;
(ii), when one adjoint polynomial of grade (n — l)o- + n — 3 vanishes in Alt ..., Au; then
* § 92. f A condition requiring in general Q<nr- l+p. t § 88.
134 SPECIAL SETS. EQUIVALENT SETS. [93
there are R — 1 of the places Blt ..., BR such that every adjoint polynomial of grade
(n — ])a- + n — 3 + r, vanishing at these places, vanishes at the remaining place. For an
example of this case we may cite the theorem : If a cubic curve be drawn through three
collinear points A^ A2, A3 of a plane quartic curve, the remaining nine intersections
BU ..., Bg are such that every cubic through a proper set of eight of them necessarily
passes through the ninth. In general any set of eight of them may be chosen.
When r + 1 is greater than zero we may take the polynomial ty itself to
be of grade (n — 1) a + n — 3. Since then a general polynomial ^ of grade
(n — 1) o- + w - 3 contains p arbitrary coefficients, we can similarly prove
that
When r + 1 adjoint polynomials of grade (n— 1) <r + ?i — 3 vanish in Q
places A1} ..,, AQ, so that the Q places have the strength Q — q as deter -
minators of adjoint polynomials of grade (n — 1) cr + n — 3, their residual
B1} ... , BR, of R = 2p — 2 — Q places, have the strength p — q — 1, or R — r, as
determinators of adjoint polynomials of grade (n — 1) cr + n — 3. In this case
the numbers are connected by the equations
and the characters of the sets Aly ..., AQ> Blt ..., BR are perfectly reciprocal*.
Ex. When the strength of a set Alt ... , AQ, wherein Q<p, as determinators of adjoint
polynomials of grade (n— l)<r + »-3, is equal to their number, so that the number of
linearly independent adjoint polynomials of grade (n— l)o- + n — 3 which vanish in the
places of the set is given by r + 1 =p — Q, it follows that g = 0. Thus if Blt ... , BR be the
residual zeros of an adjoint polynomial, <£, of grade (n-l)ar + n — 3, which vanishes in
Alt ..., A01 so that R + Q=2p — 2, only one adjoint polynomial of grade (»— l)<r+7&— 3
vanishes in B1} ... , JBR, namely <£.
94. It is known that the number of places *}* of the Riemann surface
at which a rational function takes an arbitrary value c, is the same as the
number of places at which the function is infinite. The sets of places at
which c has its different values, may be called equivalent sets of places for
the function under consideration. For such sets we can prove the result : —
if a set of places A^, ..., A'Q be equivalent to a set Alt ..., AQ, in the sense
that a rational function g takes the value c at each place of the former set
and at no other places, and takes the value c at each of Alf ..., AQ and
at no other places of the Riemann surface, then the general rational function
with simple poles at A^, ...,A'Q contains as many linearly entering arbitrary
constants as the general rational function whose poles are at Alt ..., AQ.
* For the theory of such reciprocal sets from the point of view of the algebraical theory of
curves, see the classical paper, Brill u. Noether, "Ueber die algebraischen Functionen u.s.w.",
Math. Annal. vii. p. 283 (1873).
t In this Article, when a rational function g is said to have the value c at a place, it is
intended that g - c is zero of the first order at the place. A place where g - c is zero of the k-th
order is regarded as arising by the coalescence of k places where g is equal to c.
95] COKESIDUAL SETS. 135
For let the general rational function with poles at A1} ..., AQ be denoted
by G, and be given by
0 = v0 + vlGl+ ...... +vqGq,
where v0, ..., vq are arbitrary constants, and Glt ... , Gq are particular functions
whose poles are among Al} ..., AQ — of which one, say Glt may be taken
to be the function (g — c')/(g — c). Then if G' denote any function what
ever having poles AI, ..., A'Q, and not elsewhere infinite, the function
G' (g — c')/(g — c) is one whose poles are at Al} . . . , AQ ; thus G' (g — c')/(g — c)
can be expressed in the form
for proper values of v0, ..., vq. Therefore G' can be expressed in the form
n> q — c „ q — c ~ q — c
'
Since this is true of every function whose poles are at A^ , ..., A'Q, and that
the functions G%(g — c)j(g — c) are functions whose poles are at AI , . .., A'Q,
the result is obvious.
95. If the symbol GO be used to denote the number of values of an
arbitrary (real or complex) constant, the general adjoint polynomial ^, of
grade (n — 1) cr -+ n — 3 + r, of the form
which vanishes in the places B1} . .., BR, gives rise to oo q sets of places,
constituted by the zeros of S- other than Blt ...,BR, each set consisting of,
say, Q places. Let Al} ..., AQ be one of these sets.
We shall say that these sets are a lot of sets ; that each set is a residual
of Blt ..., BR, and that they are co-residual with one another; in particular
they are all co-residual with the set Alt ..., AQ. Further we shall say that
the multiplicity of the sets, or of the lot, is q, and that each set has the
sequence Q — q ; in fact an individual set is determined by q independent
linear conditions, namely, of the Q places of a set, q can be prescribed and
the remaining Q — q are sequent.
It is clear then that any set, AI, ..., A'Q, which is co-residual with
Alt..., AQ, is equivalent with Alf...,AQ, in the sense of the last article;
for these two sets are respectively the zeros and poles of the same rational
function ; in fact if i/r be the polynomial vanishing in Bit ..., BR> Alf ..., AQ,
and ^ the polynomial vanishing in Blt ..., BR, AI, ..., A'Q, the rational
function ^/i/r has J./, ..., A'Q for zeros and Alf ..., AQ for poles. Hence
by the preceding article it follows that the number q + 1 of linear, arbitrary,
coefficients in a general rational function prescribed to have its poles at
AI, ..., AQ) is the same as the number in the general function prescribed to
136 THEOREM OF CORESIDUAL SETS. [95
have its poles at the co-residual set J./, ..., A'Q. In other words, co-residual
sets of places have the same multiplicity, this being determined by the
number of constants in the general rational function having one of these
sets as poles ; they have therefore also the same strength Q — q, or p — (T + 1),
as determinators of adjoint polynomials of grade (n — I)<r + n — 3.
96. In the determination of the sets co-residual to a given one, Alt ...,
AQ, we have made use of a particular residual, B1} ..., Blt. It can however
be shewn that this is unnecessary — and that, if two sets be co-residual for any
one common residual, they are co-residual for any residual of one of them. In
other words, let an adjoint polynomial ^r, of grade (n — l)a+n— 3 + r, be
taken to vanish in a set A1} ..., AQ, its other zeros (besides those where
y'(^) = 0( or F' (77) = 0), being Bl} ...,BR, and an adjoint polynomial ^, of
grade (n — 1) <r + n — 3 + r, be taken to vanish in Blt ..., BR, its other zeros
being the set A^, ..., A'Q, co-residual with A1} ..., AQ; then if an adjoint
polynomial, i/r', of grade (n — l)o- + n — 3 + r', which vanishes in A1} ..., AQ,
have BI, ..., B'R> for its residual zeros, R being equal to nr' + 2p — 2 — Q, it
is possible to find an adjoint polynomial ^', of grade (n — l)cr + n — 3 + r,
whose zeros are the places B^, ..., B'R,, A^, ..., A'Q.
For we have shewn that any rational function having Alf ..., AQ as its
poles can be written as the quotient of two adjoint polynomials, of which the
denominator is arbitrary save that it must vanish in the poles of the function,
and be of sufficiently high grade to allow this. In particular therefore the
function ^f/^r, whose zeros are AI, ..., A'Q, can be written as the quotient of
two polynomials of which ijr' is the denominator, namely in the form *$*' Itf .
The polynomial ^' will therefore vanish in the places J3/> ..., B'K, A^, ...,A'Q,
as stated.
Hence, not only are equivalent sets necessarily co-residual, but co-residual
sets are necessarily equivalent, independently of their residual*.
97. The equivalence of the representations ^/\^, ^'ty\ nere obtained, of the same
function, has place algebraically in virtue of an identity of the form
where /=0 is the equation associated with the Riemann surface and K is an integral poly
nomial in x and y. Reverting to the phraseology of the theory of plane curves, it can in
fact be shewn that if three curves /= 0, ^ = 0, H= 0 be so related that, at every common
point of / and •<//-, which is a multiple point of order k for / and of order I for ^ , whereat
/and ^ intersect in H + /3 points, the curve ^Thave a multiple point of order k + l-l+ft,
so that in particular If passes through every simple intersection of / and ^r, then there
exist curves ^' = 0, K=0, such that, identically,
Now in the case under consideration in the text, if the only multiple points of / be
multiple points at which all the tangents are distinct, the adjointness of ^ ensures that ^
* For the theory of co-residual sets for a plane cubic curve see Salmon, Higher Plane Curves
(Dublin, 1879), p. 137. That theory is ascribed to Sylvester; cf. Math. AnnaL, t. vii., p. 272 note.
98] FUNCTIONS EXPRESSIBLE BY ^-POLYNOMIALS. 137
has a multiple point of order is— I at every multiple point of/ of order k. The adjointness
of the polynomials ^, ^' ensures that the compound curve ^' has a multiple point of
order 2 (k- 1) or Ic + k - 1 - 1 at every multiple point of / of order L Further, the curve
^^' passes through the simple intersections of / and \^, which consist of the sets
AI,...,AQ, Si, ...,BR; for ^ passes through Bl,...,BR, and ^' is drawn through
AI, ... , AQ. Hence the conditions are fully satisfied in this case by taking 11=^^' ; thus
there is an equation of the form
from which it follows that the curve ^' is adjoint at the multiple points of / and passes
through the remaining intersections of / and ^^', namely through A\, ...,A'B and
B\ , ... , B'tt. . This is the result of the text.
In case of greater complication in the multiple points of/ there is need for more care
in the application of the theorem here quoted from the algebraic theory of plane curves.
But this theorem is of great importance. For further information in regard to it the
reader may consult Cayley, Collected Works, Vol. i. p. 26 ; Noether, Math. Annal. vi.
p. 351 ; Noether, Math. Annal. xxiii. p. 311 ; Noether, Math. Annal. xl. p. 140 ; Brill and
Noether, Math. Annal. vii. p. 269. Also papers by Noether, Voss, Bertini, Brill, Baker in
the Math. Annal. xvii, xxvii, xxxiv, xxxix, xlii respectively. See also Grassmann, Die
Ausdehnungslehre von 1844 (Leipzig, 1878), p. 225. Chasles, Compt. Rendus, xli. (1853).
de Jonquieres, Me'm. par divers savants, xvi. (1858).
98. From the theorem, that a lot of co-residual sets, of Q places, may be
regarded as the residual of any residual of one set, SQ, of the lot, it follows,
that every lot wherein the sequence of a set is less than p, may be determined
as the residual zeros of a lot of adjoint polynomials of grade (n — 1) <r + n — 3,
which have R = 2p — 2 — Q common zeros. For the sequence Q — q is equal
to p — (T + 1), and when r+l>0 an adjoint polynomial (involving r-f-1
arbitrary coefficients) can be determined which is zero in any one set, SQ, of
the lot, and in R other places.
Hence also, when Q places are such that the most general rational
function, of which they are the poles, contains more than Q — p + 1 arbitrary
constants, this general rational function can be expressed as the quotient of
two adjoint polynomials of grade (n— l)<r + n — 3; the same is true when
the Q places are known to be zeros of an adjoint polynomial of grade
(?i- l)o- + /i-3.
It follows from what was shewn in Chap. III. §§ 23, 27, that if p places be
the poles of a rational function, an adjoint polynomial of grade (n— l)o- + w— 3
vanishes in these places ; and an adjoint polynomial of that grade can always
be chosen to vanish in p — 1, or a less number, of arbitrary places. Hence,
every rational function of order less than p 4- 1, is expressible as the quotient
of two adjoint polynomials of grade (n — 1) <r + n — 3.
Ex. i. A rational function of order 2p — 2 which contains p, or more, arbitrary constants
(one being additive) is expressible as the quotient of two adjoint polynomials of grade
Ex. ii. For a general quartic curve, co-residual sets of 4 places with multiplicity 1 are
determined by variable conies having 4 given zeros ; but co-residual sets of 4 places with
138 POSSIBLE DEPENDENCE [98
multiplicity 2 are determined as the zeros of variable polynomials of degree 1, i.e. by
straight lines.
Ex. iii. The equation of a plane quintic curve with two double points, can be written
in the form ^S' -^'S—0, where ^, ^-' are cubics passing through the double points and
seven other common points, and S, S' are conies passing through the double points and
two other common points.
Ex. iv. When r + l adjoint polynomials of grade (n— l)a- + n — 3 vanish in a set, £fl, of
Q places, there must be p — T— 1 independent places Alt ... , Ap_r,1, in Ss, such that
every adjoint polynomial of grade (n— \)<r + n — 3 which vanishes in them vanishes of
itself in the remaining q places AP_T, ... , AQ. Let SR be a residual of SB, R being equal
to 2p — 2-Q. Then, regarding SR and AP_T, ..., Ae, together, as forming a residual of
Alf ..., Ap_r_1, it follows (§ 93) that there is only one adjoint polynomial of grade
(n — I)<r+n — 3 which vanishes in SK and in AP_T, ... , AQ. Hence there exists no rational
function having poles only at the places Aly ..., AP_T_V For such a function could be
expressed as the quotient of two adjoint polynomials of grade (n — l)<r + n — 3 having
SR and Ap_r, ... , Aa as common zeros. Compare § 26, Chap. III.
It can also be shewn, in agreement with the theory given in Chapter III., that if
.Z?,, ... , JBr'+i be any r' + l independent places, T being less than T, there exists no rational
function having poles in £fi and Blt ... , BT-+I. In fact r-f 1 - (T' + 1) linearly independent
adjoint polynomials of grade (n— l)<r + n — 3 vanish in Ss and Bly ..., Br'+\. Let SR,,
where R' = 2p — 2 - (Q + r' + 1), be the residual zeros of one of these polynomials. Then the
strength of SRI, as determinators of adjoint polynomials of grade (n — l}a- + n — 3 is (§ 93)
R' — (T — r') + l = R — T, where R = 2p — 2 — Q, namely the strength of SR, is the same as the
strength of SR, and Bv,...,Br'+i together; hence every adjoint polynomial of grade
(n— l)o-+n — 3 which vanishes in SR,, vanishes also in Blt ... , Br'+i. Now every rational
function having £2 and Blt ... , Br'+i as poles, could be expressed as the quotient of two
adjoint polynomials of grade (n- l)<r+n — 3 having SR, as common zeros; since each of
these polynomials will also have Blt ... , -5r'+i as zeros, the result is clear.
99. The remaining Articles of this Chapter are devoted to developments
more intimately connected with the algebraical theory of curves.
We have seen that an individual set of a lot of co-residual sets of Q
places is determined by the prescription of a certain number, q, of the places ;
this number q being less than* Q and not )iprSAt6r than Q— p.
But it does not follow that any q places of a set are effective for this
purpose ; it may happen that q places, chosen at random, are ineffective to
give q independent conditions.
We give an example of this which leads (§ 100) to a result of some interest.
Suppose that a set of Q places, 8Q, is given, in which no adjoint polyno
mial of grade (n— l)a- + n— 3 vanishes ; then r + 1 is zero, and co-residual
sets are determined by Q— p places. Suppose that among the Q places there
are p + s — 1 places, forming a set which we shall denote by <rp+s-i , which
are common zeros of T' -f 1 adjoint polynomials of grade (n — l)cr + n — 3;
denote the other Q— p — s + 1 or q — s +-I places by <r9_g+1.
* The formula is Q -q=p- (r + 1); if q were Q and therefore r + l=p, all adjoint poly
nomials of grade (n- 1) <r + n-3 would vanish in the same Q places, contrary to what is proved
in § 21, Chap. II.
100] OF PLACES OF A SET. 139
Take an adjoint polynomial of grade (?i — 1) <r + n — 3 + r which vanishes
in the places of the set SQ, and let SR denote its remaining zeros, so that
R + Q = nr + 2p — 2. If we now regard the sets SB, <rq-t+i together as the
residual of the set <Tp+t-i, it follows (§ 93) that SK, o-q-g+l together have only
the strength R + q — s + 1 — (T + 1), or nr +p — 2 — (T + s), as determinators
of polynomials of grade (n — l)cr + 7i-3 + r; and if we choose 5 — 1 places
Alt ..., Ag_! from o-p+g-i, the polynomial of grade (n — 1) & + n — 3 + r with
zeros in SR, which vanishes in the q places constituted by a-q_s+1 and
A!, ..., Ag..! together, will not be entirely determined, but will contain*
T' + 2 arbitrary coefficients, at least-}- : thus r' + 1 further zeros must be
prescribed to make the polynomial determinate.
A particular case of this result is as follows : — Consider a lot of co-
residual sets of Q, = q + p, places, in which no adjoint polynomial of grade
(n — l)<r + n — 3 vanishes. If p of the places of a set be zeros of r'+l
adjoint polynomials of grade (n— l)<r + w — 3, then the other q places are not
sufficient to individualise the set ; r + 1 additional places are necessary.
For instance a particular set from the double infinity of sets of 5 places, on a plane
quartic curve, determined by variable cubic curves having seven fixed zeros, is generally
determined by prescribing 2 places of the set. But if there be one of the sets for which
3 of the five places are collinear, then the other two places do not determine this set ;
we require also to specify one of the three collinear places. It is easy to verify this result
in an elementary way.
100. Consider now two sets 8R, SQi, which are residual zeros of an
adjoint polynomial, i/^, of grade (n— 1) a + n — 3 + rl} so that
Let Xr_Tl + l be the number of terms in the general non-adjoint polynomial
of grade r — 1\ and Nr^fi be the total number of zeros of such a non-adjoint
polynomial of grade r — r^ Take Xr_ri independent places on the Riemann
surface, forming a set which we shall denote by Tr_ri, and determine a non-
adjoint polynomial, ^, of grade r — rly to vanish in Tr_ri. It will vanish in
Nr_ri — Xr_ri other places, Ur-ri. Suppose that no adjoint polynomials of
grade (n— l)a + n — 3 vanish in all the places of SQt and Tr_n. The product
of the polynomials fa and % is an adjoint polynomial of grade (n — l)a + n
— 3 + r. A general adjoint polynomial of grade (n — ~L) <r + n — 3 + r which
vanishes in SR will vanish in all the places forming SQl, Tr_ri, Ur_ri together,
provided we choose the polynomial to have a sufficient number of these
places as zeros. Divide the set SQl into two parts, one, T, consisting of
Qi ~P + C^r-r, - -XV-r,) places, the other U consisting of p - (Nr_ri - Xr_ri)
* For nr+p-2 is the number of independent zeros necessary to determine an adjoint poly
nomial of grade (n - l)<r + n - 3 + r.
t More if the 8 -I places Av ..., At_l be not independent of the others already chosen.
140 CAYLEY'S THEOREM. [100
places. The sets T and Tr_fi together consist of Qi — p + Nr_ri, or Q — p,
places, where
Q = Qi + Nr-rt . = nr + 2p—2 — R,
for Nr_Ti = n (r — r^, (§ 86, Ex. iii.); if then the sets U and Ur-r^ together
are not zeros of any adjoint polynomial of grade (n— l)<r + n — 3, the general
adjoint polynomial, of grade (n — 1) cr + n — 3 + r, which vanishes in SR, will
be entirely determined by the condition of vanishing also in the places of
T and Tr^fl, and will of itself vanish in the remaining places U and Z7r_ri.
If, however, r'+l adjoint polynomials of grade (n — 1) a + n — 3 — (r — rx)
vanish in the places U, the products of these with the non-adjoint polynomial
^ give -r'+l adjoint polynomials of grade (n — 1) a + n— 3 vanishing in U
and Z7r_ri. In that case, assuming that no adjoint polynomials of grade
(n— l)a- + n — 3 vanish in the p places U, Ur-rj, other than those contain
ing % as a factor, the adjoint polynomial of grade (n— l)<r + n — 3 + r which
vanishes in SK, T and Tr_ri, will require T' + 1 further zeros for its complete
determination (§ 99).
Since now the set Tr-Tl entirely determines the set Ur-ri> we may drop
the consideration of it, and obtain the result —
The adjoint polynomial, of grade (n— l)cr + n — 3 + r, which vanishes in
all but p — (Nr-ri — Xr-ri) of the zeros of an adjoint polynomial of grade
(n — 1) cr + n — 3 + rlt will have a multiplicity T' + 1 + Xr_ri , where r' + l is
the number of adjoint polynomials of grade (n — 1) <r 4 -n — 3 — (r — i\) which
vanish in these other p — Nr-ri + Xr_ri zeros. When r' + l is zero the adjoint
polynomial of grade (n — 1) cr + n — 3 + r vanishes of itself in the remaining
p — Nr-ri+Xr_ri zeros of the adjoint polynomial of grade (n — l)cr + n — 3+rj.
When r' + l is not zero it is necessary, for this, to prescribe T' + 1 further
places of these p — Nr-r, + Xr_ri zeros (provided r' + 1 < p — Nr-fl + Xr_Ti).
We have noticed (§ 8G, Ex. iii.) that
Nr_ri = n(r- n),
where E (x) denotes the greatest integer in x.
For <r = 0, therefore, the number p - Nr_Tl + Xr_Tl is immediately seen to
be equal to
where 7 = n - (r - r^, and £/ is the sum of the indices, of the surface, for
finite and infinite places (§ 88).
Thus the result, for o- = 0,— an adjoint polynomial of degree n — 3 + r
which vanishes in all but £ (7 - 1) (7 - 2) - \I of the zeros of an adjoint
polynomial of degree n — 3 + r^ (r > rl5 7 = n - (r - r,) <fc 3) will have a
101] CAYLEY'S THEOREM. 141
multiplicity r + 1 -f £ (n — 7) (n — 7 + 3), where T + \ is the number of adjoint
polynomials of degree 7-8 which vanish in the £(7 — 1) (7 — 2)- |7 wn-
assigned zeros ; if r + 1 is zero this polynomial of degree n — 3 + r will of
itself vanish in these unassigned zeros : if r + 1 > 0 it is necessary, for this, to
prescribe r' + 1 or, if r + 1 > \ (y — 1) (7 — 2) — £7, to prescribe all the un
assigned zeros.
For example let n = 5 ; take as the fundamental curve a plane quintic with 2 double
points (p = 4) ; let the remaining point of intersection with the quintic, of the straight line
drawn through these double points, be denoted by A.
(i) Take r=2, rt = l. Then y = 5-l = 4, y-3 = l; thus, an adjoint quartic curve
vanishing in all but £(y — l)(y — 2) — 2, or 1, of the zeros of an adjoint cubic, that is,
vanishing in 10 of these zeros, beside vanishing at the double points, will have a multi
plicity T' + 1 + £4, or T + 1 + 2, where r + 1 is zero if the non-assigned zero be not the point
A : and this quartic will then, of itself, pass through the unassigned zero. In this case, in
fact, the prescription of the 10 + 2 zeros of the quartic on the cubic, is a prescription of
more than 4.3-jOj, where pl is the deficiency of the cubic. Hence the quartic will
contain the cubic wholly, as part of itself. (In general, the condition to provide against
this can be seen to be r > 3.)
(ii) Take the same fundamental quintic, with r = 4, ^ = 3. Then an adjoint sextic
curve, •«//•, passing through all but £3 . 2 — 2, or 1, of the zeros of an adjoint quintic, ^, that
is through 20 of them, will have multiplicity r' + l-j-2, where r'+l is zero unless the other
zero of the quintic, ^, be the point A.
If however the unassigned zero of the quintic, ^-, be the point A, the 20 points are not
sufficient ; the sextic, ^, has multiplicity 3 and the 20 points plus A are necessary to
make ^ go through the remaining 7 points.
It should be noticed that an adjoint curve of degree 7 — 8 can always be
made to pass through ^ (7 — 1)(7 — 2) — ^1 — 1 places. The peculiarity in
the case considered is that such curves pass through one more place.
The theorem here proved was first given by Cay ley in 1843 (Collected Works, Vol. i.
p. 25) without special reference to adjoint curves. A further restriction was added by
Bacharach (Math. Annal. t. 26, p. 275 (1886)).
101. In the following articles of this chapter we shall speak of an
adjoint polynomial of grade (n — 1) <r + n — 3 as a ^-polynomial. In chapter
III. (§ 23) we have seen that the set of places constituted by the poles
of a rational function, is such that one of them ' depends ' upon the others ;
thus (§ 27) there is one place of the set such that every ^-polynomial vanish
ing in the other places, vanishes also in this. Conversely when a set of
places is such that every (^-polynomial vanishing in all but one of the places,
vanishes of necessity also in the remaining place, this remaining place
depends upon the others*. When a set S is such that every ^-polynomial
* Or on some of them. For instance, if in a two-sheeted hyperelliptic surface, associated with
the equation y*=(x, l)2P+2> we *a^e three places (arlt yj, (x.2, t/2), (o^, -y2), every <f>- polynomial,
(ar-a;,) (x-o;2) (x, l)p_3, of order p- 1 in x, which vanishes in (a:,, ?/j), (a;2, j/2), vanishes also in
(x2, -i/a). But this last place does not, strictly, 'depend' on (xlt j/j) and (ar2, y2); it depends on
(x2, j/2) only.
142 TRANSFORMATION OF FUNDAMENTAL EQUATION [101
vanishing in S, vanishes also in places A , B, . . . , it will be convenient, here, to
say that these places are determined by S.
Take now any p — 3 places of the surface, which we suppose chosen
in order in such a way that no one of them is determined by those preceding.
Then the general ^-polynomial vanishing in them will be of the form
\<f> -f fjfe + v^r, wherein A,, /JL, v are arbitrary constants and <j>, ^, ty are
^-polynomials vanishing in the p — 3 places. We desire now to find a
place (ajj) such that all (^-polynomials vanishing in the p — 3 given places
and in xlt shall vanish in another place ac». For this it is sufficient that
the ratios <f> (X) : S- (#j) : t/r (o^) be equal to the ratios <£ (#2) : ^ (#2) : ^ (#2)-
From the two equations thus expressed, with help of the fundamental
equation of the surface, we can eliminate x2, and obtain an equation for xlt so
that the problem is in general a determinate one and has a finite number of
solutions : as a matter of fact (§ 102, p. 144, § 107) the number of positions
for xl is zp(p — 3)*, and each determines the corresponding position of #2.
Hence there exist on the Riemann surface oo p~3 sets of p — 1 places such
that a single infinity of ^-polynomials vanish in them ; such a set can be
determined from p — 3 quite arbitrarily chosen places, and, from them, in
\P (P ~ 3) ways. Putting Q = p — 1, T + 1 = 2, we obtain, by the Riemann-
Roch Theorem q = 1 . Hence to each set once obtained there corresponds
a single infinity of co-residual sets.
102. The reasoning employed in the last article, to prove that there
are a finite number of positions possible for #1}and the reasoning subsequently
to be given to determine the number of these positions, is of a kind that
may be fallacious for special forms of the fundamental equation associated
with the Riemann surface. An extreme case is when the surface is hyper-
elliptic, in which case all the ^-polynomials vanishing in any given place
have another common zero (Chap. V. § 52). In what follows we consider only
surfaces which are of perfectly general character for the deficiency assigned.
In particular we assume, what is in accordance with the reasoning of the
last article, that not every set of p — 2 places is such that the two (or more)
linearly independent ^-polynomials vanishing in them, have another common
zero*f*.
* This result is given in Clebsch and Gordan, Theorie der Abel. Funct. (Leipzig, 1866) p. 213.
t Noether (Math. Annal. xvii.) gives a proof that this is true for every surface which is not
hyperelliptic. Take a set of p - 2 independent places, denoted, say, by S, and, if every p- 2 places
determine another place, let A be the place determined by the set S. Take a further quite
arbitrary place, B. When the surface is not hyperelliptic, B will not determine another place.
Each of the \ (p - 1) (p - 2) sets, of p - 3 places, which can be selected from the p - 1 places formed
by S and A, constitutes, with B, a set of p-2 places, and, in accordance with the hypothesis
allowed, each of these sets determines another place. It is assumed that the p-2 places S, and
the place B, can be so chosen that the J (p - 1) (p - 2) other places, thus determined, are different
from each other and from the p places constituted by S, A and B together. Since the places S are
independent, the ^-polynomial vanishing in S and B is unique; and, by what we have proved,
102] BY (^-POLYNOMIALS. 143
Then it will be possible to choose p — 3 independent places, S, as in the
last article, such that there is a finite number of solutions of the problem of
finding a place (#j) such that the ^-polynomials vanishing in S and (#j), have
another common zero ; let p — 3 places, forming a set denoted by S, be
so chosen. Let A be a place not coinciding with any of the positions possible
for a?,, and not determined by S. Let <f>, ^ be two linearly independent
(^-polynomials vanishing in S and A. Then the general ^-polynomial vanish
ing in S and A is of the form \(f> -I- fjfo, \ and /t being arbitrary constants,
and the general ^-polynomial vanishing in the places S only can be written
in a form \(j> + /j$r + vfy, wherein v is an arbitrary constant and ty is a
(^-polynomial so chosen as not to vanish at the place A.
Consider now the rational functions* z=(f)/ty, s.=^f/ty, each of the
(p + l)th order. They both vanish at the place A.
These functions will be connected by a rational algebraic equation,
(s, z) = 0, obtained by eliminating (x, y} between the fundamental equation
and the equations zty = <f>, sty = ^ ; associated with the equation (s, z) = 0
will be a new Riemann surface ; to every place (#, y) of the old surface
will belong a definite place z = <f>/ty, s = *&/ty, of the new surface ; to every
place of the new surface will belong one or more places of the original surface,
the number being the same for every place of the new surface f; since there
is only one place of the old surface at which both z and s are zero, namely
the place which was denoted by A, it follows that there is only one place of
the old surface corresponding to any place of the new surface. Hence each
of x, y can be expressed as rational functions of s, z, the expression being
obtained from the equations zty = <f>, sty = ^, (s, 2) = 0 j.
Since a linear function, Az + jiS + y, equal to (X$ + ^ + 1^)/^, vanishes* at the variable
zeros of the polynomial X0 + /i^ + i/^, namely in p + l places, it follows that the equation
(s, 0) = 0 may be interpreted as the equation of a plane curve of order p + l ; the number
it vanishes in p + $ (p - 1) (p - 2) places. This number, however, is greater than 2p - 2 when p > 3.
Hence the hypothesis, that every p-2 places determine another is invalid. In case p = B the
surface is clearly hyperelliptic when every p - 2 places determine another. In case p = 2 or 1 the
surface is always hyperelliptic. It may be remarked that when we are once assured of the
existence of a rational function of p poles, we can infer the existence of a set of p - 2 places
which do not determine another (cf. § 103). We have already shewn (Chap. III. § 31) that in
general a rational function of order p does exist. The reader may prove that for a hyperelliptic
surface whose deficiency is an odd number there does not exist any rational function of order p.
* It must be borne in mind that, in dealing with a rational function expressed as a ratio of
two adjoint polynomials, we speak of its poles as all given by the zeros of the denominator; some
of these may be at x = x> (cf. § 86), and in that case their existence is to be shewn by considering
(§ 84), instead of the polynomial, \f/, of grade /*, the polynomial in •>] and £, given by £/* \f/. Or we
may use homogeneous variables (§ 85). For instance, forp-S, we may, in the text, have (§ 92,
Ex. i.) <f>=x, $=y, \l/ = l. Then 0:£:^=l:ij: £=«:«:«; and \f/ has a zero at z = oo.
t Chap. I. § 4.
J Or by the direct process of § 5, Chap. I.
144 THE 3p — 3 IRREMOVEABLE CONSTANTS. [102
of its double points will, therefore*, be \p (p-V)-p, or \p (p - 3), though it is not shewn
here that they occur as simple double points. These double points are the transforma
tions of the pairs of places, (x^, (#2), on the old surface, which were such that every
^-polynomial, vanishing in the p — 3 fixed places S, and in xlt also vanished in A'2.
Since a double point of a curve requires one condition among its coefficients, and the
number of coefficients that can be introduced or destroyed, in the equation of a curve, by
general linear transformation of the coordinates is 8, it follows that a curve of order m has
constants which are not removeable by linear transformation. In the case under con
sideration here, there are p — 3 places, £, of each of which an infinite number of positions
is possible, independently of the others, and the most general linear transformation of
* and z is equivalent only to adopting three new linear functions of </>, ^, \^, instead of
$> ^, \^, in order to express the general ^-polynomial through the places S. Hence
there are, in the new surface (s, z) effectively
that is, 3/> — 3 intrinsic constants : this is in agreement with a result previously obtained
(Chap. I. § 7).
103. The p — 3 places S may be defined in a particular way, thus : —
In general there are (Chap. III. § 31) (p — l)p(p + 1) places of the original
surface, for each of which a rational function can be found, infinite only
at such place and infinite to the pih order. Every rational function, whose
order is less than p + 1, can be expressed as the quotient of two (^-polynomials
(§ 98). The (^-polynomial, (f>, occurring in the denominator of the function,
willf vanish p times at the place where the function has a pole of order pi,
and will vanish in p — 2 other places forming a set T. The general
^-polynomial § through these p — 2 places T will not have another fixed
zero, or it would be impossible to form a rational function of order p with (f>
as denominator. Let now A denote any place of the set T, the remaining
p — 3 places being denoted by S. Then we may continue the process exactly
as in the last Article.
The p variable zeros of the ^-polynomials, of the form \<£ + /j$t, which
vanish in the p — 2 places T will, for the transformed curve, become the
variable intersections of it with the straight lines, \z + fis = 0, which pass
through the place s = 0, z = 0. We enquire now how many of these straight
lines will touch the new curve. This number may be found either by the
ordinary methods of analytical geometry || or as the number of places where
* By the formula p = %(n - 1) (no- + n - 2) - £ Si, for it is clear that s is an integral function of z
of dimension 1, so that o- = 0. And we have remarked that i is 1 at each of the places cor
responding to a double point of the curve, so that 5 + /c=4Si ; cf. Forsyth, Theory of Functions,
§182.
t See the note ( *) of § 102.
J This is the fact expressed by the vanishing of the determinant A in § 31, Chap. III.
§ Which we assume to be of the form X0 + (j&, involving q + 1 = 2 arbitrary coefficients. If q
were greater than unity, it would be possible to construct a function of lower than the yth
order. This possibility is considered below (§ 105 ff.).
|| See for example Salmon's Higher Plane Curves.
'
UN.;
103] THESE CONSTANTS DETERMINE THE EQUATION. 14)5
the differential of the function ^/(f>, of order p, vanishes to the second order,
namely* 2p + 2p—2. Among these tangents, however, there is one which
touches the transformed curve in p points, counting as p— 1 tangents.
There are, therefore, 3p — 1 other tangents. Of the 3^ distinct tangent
lines thus obtained, there are 3p — 3 distinct cross ratios, formed from the
3p — 3 distinct sets of four of them, and these cross ratios are independent of
any linear transformation of the coordinates s and z.
There are thus 3p — 3 quantities obtainable for the transformed curve.
We prove, now-f, that they entirely determine this curve, and may, therefore,
since the transformation is reversible, be regarded as the absolute constants
of the original curve. For take any arbitrary point 0 ; draw through it
3 arbitrary straight lines and draw 3p — 3 other straight lines which form
with the 3 straight lines first drawn pencils of given cross ratios. Then the
coefficients of a curve of order p + I, which passes through 0, has %p(p — 3)
double points, and touches 3p straight lines through 0, one of them in p
consecutive points, are subject to 1 + %p(p - 3) + 3p — l+p — 1 or ^p^+^p— 1
linear conditions. The number of these coefficients is ^(p+I)(p + 4<) or
zP* + §P + 2- Hence there are three coefficients left arbitrary ; besides these
there are five other constants in the equation of the curve, namely, those
which settle the position of 0 and the three arbitrary straight lines through
0. The eight constants thus involved in the curve can be disposed of by
a linear transformation.
The reader will recognise here a verification of the argument sketched in
§ 7, Chap. I. ; the present argument is in fact only a particular case of that,
obtained by specialising the dependent variable of the new surface, and the
order of the independent variable g. The restriction that the p poles of g
shall be in one place can be removed, with a certain loss of definiteness and
conviction.
The argument employed clearly fails for the hyperelliptic case, since
then the p — 2 fixed zeros of the polynomials <£ and S- determine other places,
and the function ^/<£ is not of the pih order.
Forp=3 we have the result : — If an inflexional tangent of a plane quartic curve meet
the curve again in 0, eight other tangents to the curve can be drawn from 0. The cross
ratios of the six independent sets of four tangents, which can be formed from these nine
tangents, determine the curve completely — save for constants which can l>e altered by
projection.
More generally, from any point 0 of the quartic, ten tangents to the curve can be
drawn. The seven cross ratios of these tangents leave, by elimination of the coordinates
of 0, six quantities from which the curve is determinate, save for quantities altered by
projection.
* Chap. I. § 6.
t Cayley, Collected Works, vol. vi. p. 6. Brill n. Noether, Math. Annal. t. vn. p. 303.
«• 10
146 SPECIAL SETS. [104
104. It is a very slight step from the process of the last Article to take
the independent variable to be g = ^/$, where ^, <f> are (^-polynomials, having
p — 2 common zeros forming a set such that a single infinity of ^-polynomials
vanish in the places of the set. And it may be convenient to take another
dependent variable.
In the process of Article 102, the fixed zeros of the polynomials used
are p — 3 in number, and a double infinity of ^-polynomials vanish in the
places of the set.
These two processes are capable of extension. If we can find a set SQ,
of Q places, in which just (T + 1 =) 3 ^-polynomials vanish, and if the places
SQ be such that these three ^-polynomials have no other common zero, while
the problem of finding a further place xl , such that the two ^-polynomials
vanishing in SQ and x^ have another common zero x2, is capable of only a
finite number of solutions, then we can extend the process of Article 102 ;
we can then, in fact, transform the surface into one of 2p — 2 — Q sheets.
The dependent variable in the new equation will be of dimension unity,
and the equation such as represents a curve of order 2p — 2 — Q. If, there
fore, we can find sets SQ in which Q > p — 3, the new surface will have a
less number of sheets, and therefore, in general, a simpler form of equation,
than the surface obtained in § 102.
Similarly, if we can find a set, SQ, which are the common zeros of
(r + 1 =) 2 ^-polynomials, say ^ and <£, we can use the function g = ^/(/>, with
a suitable other function, as independent and dependent variables respectively,
to obtain a new form of equation for which there are 2p — 2 — Q sheets : and
if we can get Q>p — 2 the new surface will be simpler than that obtained
in | 103.
105. We are thus led to enquire what are the conditions that r + l
linearly independent ^-polynomials should vanish in any Q places aly ..., aQ.
If the general (^-polynomial be written in the form \1(j>1(x)+...+\p(f)p(a;),
where X1} ..., \p are arbitrary constants, the conditions are that the Q
equations
,(ai)=0, (*'=1. 2' •••> Q)
should be equivalent to only p — r — 1 equations, for the determination of
the ratios \ : . . . : \p ; we suppose Q > p— r — 1, and further that the notation
is so chosen that the independent equations are the first p — T — 1 of them.
Then there exist Q — (p — T — 1) sets, each of p equations, of the form
<f>j (ap_r_1+a) = m^j (aj) + . . . + mp_T_l fe (fl^-j), (j = 1, 2, . . . , p)
for each value of a- from 1 to Q — (p — r — 1), the values of ml, ..., 7>ip_T_1
being, for any value of <r, the same for every value of j. The set, of p, of
105] SPECIAL SETS. 147
these equations, for which a- has any definite value, lead to T + 1 equations,
of the form
= 0,
4>P~r-l+k
arising for k = 1, 2, . . . , T + 1.
Putting q = Q — (p — T — 1), we have therefore q (r + 1) such equations*
connecting the Q places a1; ..., «Q.
It is obvious from the method of formation that these q(r + 1) equations
are in general independent ; in what follows we consider only the cases in
which they are independent and determinate. Then, taking Q— ^(T+I)
quite arbitrary places, it is possible to determine q (T + 1) other places, such
that there are r + 1 linearly independent ^-polynomials vanishing in the
total Q places.
The determination of the q(r + l) places, from the arbitrary Q — q(r + l) places, may be
conceived of as the problem of finding p — r-1 — [Q — <?(T + I)], or qr, places, T, to add to
the Q — q(r + \] arbitrary places, S, such that all ^-polynomials vanishing in the resulting
p — T—l places S, T, may have Q-(p-r — 1), or q, other common zeros. The^ — T — 1
places S, T are independent determinators of ^-polynomials.
For instance, when Q=p- 1, r + l = 2, it follows that q = l and Q — q (T +l)=p — 3, and
hence, from the theory here given, it follows that we can determine p — 1 places in which
two 0-polynomials vanish, and, of these, p — 3 places are arbitrary. The problem of
determining the other two places may be conceived of as the problem of determining
p-r-l-[Q — q (r + 1)], or one, other place, to add to the p — 3 places, such that all $-
polynomials vanishing in the resulting p - 2 places, which are independent determinators
of 0-polynomials, may have <? = ! other common zero. We have already seen reason for
believing that, when the p — 3 places are given, the other two places can be determined in
> — 3) ways.
To every set of Q places thus determined, there corresponds a co-residual
lot of sets of Q places, the multiplicity of the lot being q ; and every
corresidual set will have the same character as the original set. The number,
q, of places of a co-residual set which are arbitrary, cannot, obviously, be
greater than the number, Q-q(r+l), of the original set, which are
arbitrary. Hence, the self-consistence of the theory clearly requires that
Q-q(T + I)>q. From this, by means of the relation Q- q = p — T— 1, we
can deduce the two important results
P > (q + 1) (r + 1), Q>q+p -2-j •
These equations are necessary in order that alt ..., a^ should be the poles of a rational
function.
10—2
148 NORMAL EQUATION. [105
Putting Q — q (r + 1) = q + a, we obtain
1[ _.
From each such set SQ we can deduce, as its residuals, sets, SB, of
R, = 2p—2 — Q, places, in which q + l ^-polynomials vanish, and it is
immediately seen that
106. -If now we determine, in accordance with this theory, a set SQ in
which r + l=3 ^-polynomials vanish, it being assumed that these three
(^-polynomials have no other common zero, and determine <£, S- to be two
(^-polynomials vanishing in SQ and in one other place 0, ty being another
(^-polynomial vanishing in SQ but not in 0, then the equations z = tfr/ty,
s = S-/I/T, determine, as before, a reversible transformation of the surface, to
a new surface of which the number of sheets is R = 2p — 2 — Q, and in which
s is of dimension 1 in regard to z.
Since R > r + pr/(r + 1), the value of R is > 2 + \p. Thus writing p = STT,
or STT + 1, or 3?r + 2, according as it is a multiple of 3 or not, R is p — TT + 2
in all cases.
From R=p-7r + 2 follows Q=p-4 + 7r; thus q= Q - p + 3 = TT - 1,
and Q — q(T + l)=p + 7r — 4<-37r+3=p — 27r — l. This is the number
of places of the set SQ which may be taken arbitrarily. If this number
be equal to q = ir — \) it follows that, by taking two different sets of
Q — q(r + l), =p — 2-7T — 1, places, we get only two co-residual sets, and
for the purposes of forming the functions (fr/ty, *&/$*, one is as good as the
other. If however Q — q (r + 1) > q, we do not get co-residual sets by taking
different arbitrary sets of Q — q (r + 1) places : — and there is a disposeableness
which is expressed by the number of the arbitrary places, Q — q (r + 1 ),
which is in excess of the number, g, which determines the sets co-residual to
any given one.
Now Q — q(r+l) — q=p — 27T— 1 — TT + 1 = p - 3?r. And, in a surface
of m sheets and deficiency p, the number of constants independent of linear
transformations is 3m +p — 9 (§ 102). Hence the number of unassignable
quantities in the equation of the surface is
3 (p - TT + 2) +p - 9 - (p - 3-Tr) or 3p - 3 ;
and this is in accordance with a result previously obtained (§ 7, Chap. I.).
Ex. i. The values of TT for p = 4, 5 are 1, 1 respectively, and p — Tr + 2, in those cases,
= 5, 6 respectively.
Hence a quintic curve with two double points (jo = 4), can be transformed into a
quintic ; this will also have two double points, in general, since the deficiency must be
unaltered. We determine a set consisting of Q, =1, quite arbitrary place. Let the
107] ANOTHER FORM. 149
general conic through this place, and the two double points, be X0 + [*& + vty = 0. Then the
formulae of transformation are 2 = <£/^, s = ^/\^. As in the text, we may suppose 0, ^
to have another common point, in which ^ does not vanish.
Ex. ii. A quintic with one double point (p = 5) can be transformed into a sextic With,
in general, jr(6 — 1) (6 — 2) — 5 = 5 double points. For this we take p — 2n— 1 = 2 arbitrary
points ; if A$ + /^ + j/^ be the general conic through the two points and the double point,
the equations of transformation are z = ({)/\^, S=^/A//-.
Ex. iii. Shew that the orders p — IT + 2 of the curves obtainable by this method to
represent curves of deficiencies
^ = 6, 7, 8, 9
are respectively R = 6, 7, 8, 8.
107. But, as remarked (§ 104), we can also make use of sets of R places
for which T + 1 = 2, to obtain transformations of our original surface.
We can obtain such a set by taking R — T (q + 1), or R — q — 1, arbitrary
places, and determining the remaining q -\- 1 such that q + 1 ^-polynomials
vanish in the whole set of R places.
It is proved by Brill* that the number of sets of q+l thus obtainable
from R — q — 1 arbitrary places, is
where p = ^q or ^(q + l), according as q is even or odd, and [ J denotes
For instance with R=p, q = 0, the series reduces to one term, whose value is p-1,
which is clearly right ; while, when R=p — 1, <? = !, the series reduces to
p-2
or $p(p-S), as in § 101, § 102, p. 144.
When p is even and R = \p + 1, q = ^p — 1, this series can be summed,
and is equal to
2
When p is odd and R = | (p + 1) + 1, q = \ (p — 1) — 1, the series can be
summed, and is equal to
4^ |p-2/|j(l>-3) Hi + 3).
Now let \(f> + /jfo be the general ^-polynomial vanishing in a set which is
residual to one of these sets of R places, \ and p, being arbitrary constants ;
we may transform the surface with z = ^J<f> as the new independent variable.
The new surface obtained will have R sheets. The new dependent variable
may be chosen at will, provided only the transformation be reversible.
* Math. Annal. xxxvi, pp. 354, 358, 369. See also Brill and Noether, Math. Annal. vn. p. 296.
150 RIEMANN'S NORMAL EQUATION. [107
The function /^-t-X, =/iS/<£ + X, depends on 2 + R-q-l arbitrary quantities, namely
the constants X, /* and the position of the R-q-l arbitrarily taken places. There are
2/<! + 2j0-2 places where dz is zero to the second order, namely, 2/i! + 2/?-2 places where
the curve a^ + b<p = 0 touches the fundamental curve ; there remain then
2R + 2p-2-(R-q + l), = R - 1 -p + q+ 1 + 3jo-3, = 3p-3
of the 2/Z + 2J0-2 values which z has when dz vanishes to the second order, which are
quite arbitrary. Compare § 7, Chap. I.
The least possible value of R is given by the formula R > T + pr/(r + 1).
If then p be written equal to STT, or 2?r + 1, according as p is even or odd, we
may take* R = p — TT + 1, that is \p + 1 or \ (p + 1) + 1, according as p
is even or odd.
Hence, when p is even, we can determine a single infinity of co-residual
sets of ^p + 1 places, these sets being the zeros of ^-polynomials, X<£ + fjfe,
which have ^p — 3 common zeros. To determine one of these sets of \p + 1
places, we may take one place, A, arbitrarily. The other \p places can
then be determined in 2 \p — If ^p — 1 \%p + I ways. Let two of these ways
be adopted, corresponding to one arbitrary place A ; the resulting sets of
\p + 1 places will not be co-residual ; for the sets co-residual with a given
set have a multiplicity 1, and therefore no two of these sets can have a
place common without coinciding altogether. Let the sets co-residual to
these two sets be given by A</> + yu& = 0, V</>' + //$•' = 0,(f> and </>' being chosen
so as to vanish in A : we assume that </>, <£' have no other common zero.
Then the equations z = <f>fo, s = $'/*&' will determine a reversible trans
formation, as is immediately seen in a way analogous to those already
adopted. In the new equation z and s enter to a degree \p-\-\, and, since
there exists* no rational function of lower order than \p-\-\, no further
reduction of the degree to which z and s enter, is possible.
The new equation may be interpreted as the equation of a curve of order p + 2 : it
will have the form
(z, iynsm + (z, l)msm-l + ... + (z, l)m=0,
wherein m=
By putting z = \/z1, s=\/sl, it is reduced to the equation of a curve of order p. The
form possesses the interest that it was employed by Elemann.
Ex. Obtain the 2 sets of \p + 1 places corresponding to a given arbitrary point for a
quintic curve with two double points, and transform the equation.
108. If we have a set of R places^, for which r + 1 = 4, the co-residual
places being given by the variable zeros of ^-polynomials of the form
+ v<f>3 + i/r, we can, by writing
* Thus, for perfectly general surfaces of deficiency p, no rational function exists of order less
than 1 + ip. Cf. Forsyth, Theory of Functions, p. 460. Biemann, Gesam. Werke (1876), p. 101.
t Wherein R - r <p, or R <p + 3.
109] CORRESPONDENCE WITH CURVES IN SPACE. 151
and eliminating x, y from these three equations and the fundamental equation
associated with the Riemann surface, obtain two rational algebraic equations
connecting X, Y, Z\ these equations determine a curve in space, of order R',
for this is the number of variable zeros of the function \X + fj,Y + vZ + 1.
To a point X =• Xl, Y= F1; Z = Z^ of the curve in space, will correspond the
places of the surface, other than the fixed zeros of </>1; </>2, <£>3, ty, at which
and it is generally possible to choose </>], $2, <f>3, -ty- so that these equations
have only one solution.
The lowest order possible for the space curve is given by
If then p = 4-7T, or 4?r + 1, or 4?r + 2, or 4?r + 3, R may be taken equal
to p - TT + 3.
For instance with * p = 4, R = Q, taking a plane curve with double points at the places
#=oo, y = Q and x=0, y=°o , given by
x*f (x, y\ + xy (x, y\ + (ar, y)3 + (x, y\ + (x, y\ + A = 0,
we mayf take X01 + /i02 + //03 + ^ = X^y + ^ + j/i?/4-l ; the places residual to the variable
set of R places are, in number, 2p — 2-6, =0. Then the equations of transformation are
X=xt/, Y=x, Z=y,
and these give points (X, I7*, Z) lying on the surfaces,
X= YZ,
of which the first is a quadric and the second a cubic.
A set of R places with multiplicity r = 3 may of course also be used
to obtain a transformation to another Riemann surface. With the same
notation we may put z = <j>i/^r, s = (j>J^. It is clear that the resulting
equation, regarded as that of a plane curve, is the orthogonal projection, on
to the plane Z= 0, of the space curve just obtained.
A set of R places with multiplicity r > 3 may be used similarly to obtain
a curve of order R in space of r dimensions. Some considerations in this
connexion will be found in the concluding articles of this chapter.
109. It has already been explained that the methods of transformation
given in §§ 101 — 108 of this chapter are not intended to apply to surfaces
which are not of general character for their deficiency, and that, in particular,
hyperelliptic surfaces are excluded from consideration. We may give here a
practical method of obtaining the canonical form of a hyperelliptic surface,
* Since p must be 5: (T + 1) (q + 1), this is the first case to which the theory applies.
t It is easy to shew that this is the general adjoint polynomial of degree w-3. We may also
shew that the integrals, \xydxj f'(y), etc., are finite, or use the method given Camb. Phil. Trans.
xv. iv. p. 413, there being no finite multiple points.
152 HYPERELLIPTIC CASE. [109
whose existence has already been demonstrated (Chap. V. § 54). Suppose
first that p>l. In the hyperelliptic case every </>-polynomial vanishing
in any place A will vanish, of itself, in another place A'. Any one of these
^-polynomials will have 2p — 4 other zeros, forming a set which we shall
denote by S. Putting Q = 2 and r + 1 = p — 1 in the formula Q — q=p — r — \,
we find q = 1, so that the general ^-polynomial vanishing in the places S
will be of the form A,^ — X,^, wherein X1( X2 are arbitrary constants; in
fact these 2/> — 4 places S consist of p — 2 independent places and the other
p — 2 places determined by them, one by each. Thus a function of the
second order is given by z — fa/fa. A general adjoint polynomial of grade
(n — 1 ) cr + n - 2 will contain n+p—l terms and vanish, in all, in n + 2p — 2
places ; thus the general adjoint polynomial, of this grade, which is prescribed
to vanish in a set T of n 4- p — 3 arbitrary places, will be of the form
f*ifa + P&*i pi, /^ being arbitrary constants, and will vanish in p+ 1 other
places. We may suppose ^ so chosen that it vanishes in one of the two
zeros of fa which are not among the set S, and we shall assume that ty2
does not vanish in this place, and that fa does not vanish in the other
of these two zeros of fa. Then the functions z = fa/fa, s = ^i/^o, are
connected by a rational equation, (s, z) — 0, with which a new Riemann
surface may be associated ; to any place of the old surface there corresponds
only one place z= fa/fa, s = i/r1/ijr2, of the new surface; to the place z = 0,
s = 0 of the new surface corresponds only one place of the original surface,
and the same is therefore true of every place of the new surface. Thus
the equation (s, z) = 0 is of degree 2 in s and degree p + 1 in z. The highest
aggregate degree in s and z together, in the equation (s, z) = 0, is the same
as the number of zeros of functions of the form \z + JAS + v, for arbitrary
values of \, p, v, and therefore if the poles s be different from the poles
of z, namely, if the zeros of \Jr2 other than T, be different from the zeros
of fa2 other than S, the aggregate degree of (s, z) in s and z together will
be p + 3 ; thus the equation will be included in the form
s2a + s/3 + 7 = 0,
where a, /3, 7 are integral polynomials in z of degree p 4-1.
If we put a- = so. + £/3, this takes the form
0-2 = ia2 _ a%
which is of the canonical form in question.
Ex. A plane quartic curve with a double point (p = 2) may be regarded as generated
by the common variable zero A of (i) straight lines through the double point, vanishing
also in variable points A and A', (ii) conies through the double point and three fixed
points, vanishing also in variable points A, £, C.
When p is 1 or 0, the method given here does not apply, since then
adjoint ^-polynomials (which in general vanish in 2p — 2 variable places)
110] INVARIANT EXPRESSION OF FUNCTIONS THAT OCCUR. 153
have no variable zeros. In case p = 1 or p = 0, if /A^ +/i2\/r2 -f ^3-^3, with
Mi. fa, fa arbitrary, be the general adjoint polynomial of grade (n — 1) a + n — 2
which vanishes in n+p — 4 fixed places, fa, ty3 being chosen to have one
other common zero beside these n+p — 4 fixed places, we may use the
transformation z = ^/ijr.,, s = fa/ fa, z being a function of order p + I, and s
being a function of order p + 2. Then, since the function \z + /JLS + v vanishes
in p + 2 places, we obtain an equation of the form *
s2 (z, l)p + s (z, 1 ),,+1 + (z, l)p+2 = 0,
of which the further reduction is immediate.
Ex. For a plane quartic curve with two double points (^9 = 1) let nl\jsl + p.$z + ^3^3 be
the general conic through the double points and a further point A, \^x and \/r3 being chosen
also to vanish at any point B. Then we may use the transformation £ = Vri/V'3i S = ^r2/Vr3-
110. In the transformations which have been given we have made
frequent use of the polynomials which we have called (^-polynomials, namely
adjoint polynomials of grade (n— l)cr + n — 3. For this there is the special
reason, already referred to-f, that, in any reversible transformation of the
surface, their ratios are changed into ratios of ^-polynomials belonging to
the transformed surface ; thus any property, or function, which can be
expressed by these ^-polynomials on.ly, is invariant for all birational trans
formations. We give now some important examples of such properties.
Let the general ^-polynomial be always supposed expressed in the form
\fa + ... + \p<f>p, \i, ..., Xp being arbitrary constants. Instead of fa, ..., <f>p
we may use any p linearly independent linear functions of fa, ..., tj>p>
agreed upon beforehand. A convenient method is to take p independent
places GU ..., cp and define fa as the ^-polynomial vanishing in all of c1? ...,cp
except d ; but we shall not adhere to that convention in this place. Let any
general integral homogeneous polynomial in fa, ..., fa, of degree fi, be
denoted by <E>w or <&'&*>. This polynomial contains p(p + l)...(p+p- l)/p !
terms.
In a polynomial <l><2> there are $p(p + l) products of two of fa, ...,fa.
But these &p(p+l) products of pairs are not linearly independent. For
example in a hyperelliptic case, we can choose a function of the second order,
z, such that the ratios of p independent (^-polynomials are given by
fa : fa : ... : fa = 1 : z : z2 : ...
then there will be p — 2 identities of the form
= fa/fa = ... = fa/fa-! ,
* Further developments are given by Clebsch, Crelle, t. 64, pp. 43, 210. For this subject and
for many other matters dealt with in this Chapter, the reader may also consult Clebsch-
Lindemann-Benoist, Legons sur la Geometric (Paris 1883), t. in.
t Chap. II. § 21.
154 LINEARLY INDEPENDENT [110
whereby the number of linearly independent products of pairs of fa, ..., fa
is reduced to \ p (p + 1) — (p — 2), at most. But we can in fact shew,
whether the surface be hyperelliptic or not, that there are not more than
3 (p — 1) linearly independent products of pairs of fa, ..., fa. For consider
the 4 (p— 2) places in which any general quadratic polynomial, <3>(2), vanishes.
If fa fa be any product of two of the polynomials fa, ..., fa, the quotient
fafa/&(-} represents a rational function having no poles except such as occur
among the zeros* of <I>(21 ; there are therefore at least as many linearly
independent rational functions, with poles among the zeros of <I>(2), as there
are linearly independent products of pairs of fa, ..., fa,. But the general
rational function having its poles among the 4 (p — 1) zeros of <I>(2), contains
only 4>(p — 1)— p + 1, =3(p — 1), arbitrary constants. Hence there are not
more than this number of linearly independent pairs of fa, ..., <f>p. In
precisely the same way it follows that there are not more than (2/j, — l)(j» — 1)
linearly independent products of p, of the polynomials fa, ..., fa.
111. But it can be further shewn that in general ^ there are just
(2/4 — 1) (p — 1) linearly independent products of //, of the polynomials
fa, ..., fa', so that there are
identical relations connecting the products of p, of the polynomials fa, ..., fa.
Consider the case p, = 2. Take p - 2 places such that the general
(^-polynomial vanishing in them is of the form \fa + /j.fa, A, and p being
arbitrary, and fa, fa having no zero common beside these p - 2 places. Let
4>(D> <|>'(i) denote two general linear functions of fa, ..., fa. The polynomial
is quadratic in fa, ..., fa. It contains 2p terms. But clearly these terms
are not linearly independent, for the term $2 fa occurs both in fa<&w and
in </>2<£>'(1). Suppose, then, that there are terms, faWw, occurring in fa$>'(l},
which are equal to terms, ^>i^(1), occurring in fa$>m. The necessary equation
for this,
V® = fa
~ ~ fa '
shews that ^(1) vanishes in the p zeros of fa which are not zeros of fa.
But since these p zeros form a set which is a residual of a set (of p — 2 places)
* Here, as in all similar cases, the zeros of the polynomial are its generalised zeros when it
is regarded as of its specified grade.
t Precisely, the theorem is true when the surface is sufficiently general to allow the existence
of p-2 places such that the general 0-polynomial, vanishing in them, is of the form X^ + pfa,
\ and M being arbitrary constants, and fa, <f>.2 having no common zero other than the p - 2
places. We have already given a proof that this is always the case when the surface is not
hyperelliptic (§ 102).
Ill] PRODUCTS OF ^-POLYNOMIALS. 155
in which two (^-polynomials vanish, it follows* that only one ^-polynomial
vanishes in these p places; and such an one is fa. Hence "^(1) must be
a multiple of fa, and therefore XP'(1) a multiple of fa. Thus the polynomial
contains 2p — 1 linearly independent products of pairs of fa, ..., fa,.
Let now fa be a ^-polynomial not vanishing in the common zeros of
</>,, fa, and let fa, ..., fa be chosen so that fa, fa, fa, ..., fa are linearly
independent. Consider the polynomial
4> = fa<&n + fa& «') + fa \\3fa + . . . + \pfal
wherein \3, ..., \p are arbitrary constants. Herein fa(\^fa + ...+\pfa)
cannot contain any terms fa (X/$3 + . . . + ^p'fa) which are equal to terms
already occurring in the part fa<btl} + fa<&'ft), or else \3'<f>3 + ... + \p'fa would
vanish in the p — 2 common zeros of fa and fa ; and this is contrary to the
hypothesis that \fa + ufa is the most general ^-polynomial vanishing in
these p — 2 places. Hence the polynomial <E> contains 2p — 1 + p — 2, or
Sp — 3, independent products of twos of the polynomials fa, ...,fa. As
we have proved that a greater number does not exist, 3p — 3 is the number
of such products of pairs.
Consider next the case yu, = 3. Since co-residual sets of 2p — 1 places
have f a multiplicity p — 1, it follows that the general polynomial, \f(2), of
the second degree in fa, ..., fa, which vanishes in 2p — 3 fixed places, and
therefore in 2p — I variable places, contains p arbitrary coefficients. If then
the 2p— 3 fixed zeros of ^(2) be zeros of a definite polynomial, fa, it follows
that ¥<-> is of the form fa^f^, ^ being of the first degree in fa, fa, ..., fa.
Hence, as in the case /u- = 2, it can be proved that if fa, fa be ^-polynomials
with one common zero, the reduction in the number, 2('3p — 3), of terms
in a polynomial </>!<!> (2> + fa&{2), which arises in consequence of the occurrence
of terms, faW®, in fa<&'(2), which are equal to terms, - faW®, occurring
in fa<&{2). is at most equal to p. Hence the polynomial fa<&® + fa<&'{2)
contains at least 5p — 6 linearly independent products of threes of fa, ..., fa.
Hence taking fa, and a quadratic polynomial 3>"<2», such as do not vanish
in the common zero of fa, fa, it follows that a cubic polynomial with at least
op — 5 linearly independent products, is given by
We have thus proved that in the cases /* = 2, /A = 3, the polynomial
contains (2/u,- l)(p — 1) linearly independent products. Assume now
that 4>^-») contains (2/*- 3) (p- 1) independent terms, and that <£<"-2»
* From the formula (Chap. VI. § 93)
Q-R = 2(q-r),
putting Q=jp-2, R=p, r=l, we obtain q = 0.
t From Q-q=p-(r + l), putting r+l=0 (because 2p- l>2p-2) Q = 2p-l, q=p-l.
156 THE ^-POLYNOMIALS FORM A COMPLETE SYSTEM. [Ill
contains (2/x — 5)(p — 1) independent terms. A general polynomial
vanishing in the zeros of a definite ^-polynomial, <£2, will have 2(/i — 2)(j9 — 1)
variable zeros; and the multiplicity of co-residual sets of 2 (/A — 2)(p — 1)
places, when /x > 3, is (2/j, — 5)(p — 1) — 1, which by hypothesis is the same
as the multiplicity of the sets of zeros of a polynomial <$>,?¥ (>i~z), in which
-2) nas i£s most general form possible. Hence the general polynomial
vanishing in the zeros of <£2, is of the form <f>^^~2). If then, in a
polynomial, (j)^^"^ + <f>2<&' (*~v , of the yu-th degree in </>l5 ..., <f)p, wherein
<j)1} $z have no common zeros, there be terms, fy.W '(ft~1) , occurring in <£2<l>/('1~1),
which are equal to terms, - <£1Mr('t~1), occurring in ^>1<I>('t~1), then ty^-v must
be of the form ^F^-2*, and ^V-1' of the form c^'^-2', and the resulting
reduction in the number, 2 (2/i - 3)(p - 1), of terms in ^tl?"*-1* + ^2<l>/"t-1),
is at most equal to the number, (2//, — 5) (p — I), of terms in a polynomial
^c*-2*. Thus, there are at least
2 (2,1 - 3) G>-1)- (2,1-5X0-1), = (^-l)(p-l\
linearly independent terms in the polynomial ^^^"^ + $.,&>' ^~l) ; as we have
proved that no greater number exists, it follows that (2/i — l)(p— 1) is the
number of linearly independent products of yu, of the polynomials <f>1} ..., (f>p.
112. Another most important theorem follows from the results just
obtained : Every rational function whose poles are among the zeros of a
polynomial ^w can be expressed in a form (pM/tyM. For the most general
function having poles in these 2/u, (p — 1) places contains 2/x(p — 1) — p+ 1
arbitrary constants*, and we have shewn that a polynomial <&w contains just
this number of terms; thus the quotient ^**>/¥w, which clearly has its
poles in the assigned places, is of sufficiently general character to represent
any such function.
For further information on the matter here discussed the reader may consult Noether,
Math. Annal. t. xvn. p. 263, " Ueber die invariante Darstellung algebraischer Func-
tionen." And+ ibid. t. xxvi. p. 143, "Ueber die Normalcurven fiirjo = 5, 6, 7."
In order to explain the need for the theorem just obtained, we may consider the simple
case where the fundamental equation is that of a general plane quartic curve, f(x, y, z) = 0,
homogeneous coordinates being used. If we take the four polynomials,
^i = ^2> ^2=#2> ^s=^> ^h=^>
which are not ^-polynomials, from which we obtain
x : y : z = ^ : ^3 : ^4,
* When ^>1. The theorem has already been proved for ^i = l (§ 98, Chap. VI.).
t In the present chapter all the polynomials considered in connexion with the fundamental
equation have been adjoint; there is also a geometrical theory for polynomials of any grade in
extension of the theory here given, in which the associated polynomials are not adjoint. For its
connexion with the theory here, the reader may compare Klein, "Abel. Functionen," Math.
Annal. t. 36, p. 60, Clebsch-Lindemann-Benoist, Lemons sur la Geometric, Paris 1883, t. m., also
Lindemann, Untersuchungen iiber den Riemtmn-Roch'' 'schen Satz (Teubner 1879), pp. 10, 30 etc.,
Noether, Math. Annal. t. 15, p. 507, "Ueber die Schnittpunktssysteme einer algebraischen
Curve mit nicht adjungirten Curven."
113] NOETHER'S NORMAL CURVE IN HIGHER SPACE. 157
then the general rational function with poles at the sixteen zeros of a polynomial, ¥<2>, of
the second order in fa, fa, fa, fa, contains 14 homogeneously entering arbitrary con
stants. Now there are only ten terms in the general polynomial *<2), of the second order
in fa, ... , ^4 ; and these are equivalent to only nine linearly independent terms, because
of the relation fafa = fa*. Hence the rational function in question cannot be expressed in
the form
113. The investigations in regard to the ^-polynomials fa, ..., fa, which
have been referred to in §§ 110 — 112, find their proper place in the con
sideration of the theory of algebraic curves in space of higher than two
dimensions.
Let fa, ..., fa be linearly independent adjoint polynomials of grade
(n - 1) a + n - 3, defined, suppose, by the invariant condition that if
clt ...,cp be p independent places on the Riemann surface, fa vanishes in
all of d, ..., cp except a. Let xlt ..., xp be quantities whose ratios are
defined by the equations
a?i :x2 : ... : xp = fa : fa : ... : fa.
We may suppose * that there is no place of the original surface at which
all of a:,, ..., xp are zero, and, since only the ratios of these quantities are
defined, we may suppose that none of them become infinite.
Hence we may interpret #,, ..., xp as the homogeneous coordinates
of a point in space of p - 1 dimensions ; we may call this the point as.
Corresponding then to the one-dimensionality constituted by the original
Riemann surface, we shall have a curve, in space of p - I dimensions. Its
order, measured by the number of zeros of a general linear function
A^ + ... + \yxp, will be 2/> - 2. To any place x of this curve there cannot
correspond two places c, c' of the original surface, unless
fa(c) : fa(c) : ... : fa (c) = fa (c') : fa(c') : ... : <^(c').
Now, from these equations we can infer that the (^-polynomials corre
sponding to the normal integrals of the first kind, have the same mutual
ratios at c as at c' ; such a possibility, however, necessitates the existence of
a rational function of the second order, expressible in the form
MY — /*r*/,
where X, p are constants whose ratio is definite, and T*', rj are normal
elementary integrals of the second kind with unassigned zeros. Hence the
correspondence between the original Riemann surface and the space curve,
6^-2, is reversible except in the hyperelliptic case.
In the hyperelliptic case the equations of transformation are reducible to
a form
xl : #2 : . . . : xp = 1 : z : z* : . . . : zP~\
* Chap. II. § 21.
158 NOETHER'S NORMAL CURVE. [113
To any point x of the space curve corresponds, therefore, not only the place (s, z) of the
Riemann surface, but equally the place ( - s, z). The space curve may be regarded as a
doubled curve of order p — l. (Of. Klein, Varies, ub. d. Theorie der ellip. Modulfunctionen,
Leipzig, 1890, t. I. p. 569.)
For the general case in which p = 3, the curve, C2p_2, is the ordinary
plane quartic curve. For the general case, p = 4, the curve (7^-2 is a sextic
curve in space of three dimensions, lying* on \p(p 4- 1) — (3p — 3), = 1,
surface of the second order and %p(p + l)(p + 2) — (op — 5), = 5, linearly
independent surfaces of the third order.
Ex. If, for the case jo = 4, we suppose the original surface to be associated with the
equation f
f(x,y} = tff (Lx + My) + xy (ax* + Ihxy + by2) + Px3 + Qafiy + Rxy*
+ Sy3 + Ax2 + ZHxy + Bf + Cx + Dy + 1 = 0,
and put Z—xy, X=x, Y=y, as the non-homogeneous coordinates of the points of the
curve C"2p_2, the single quadric surface containing the curve is clearly given by
U2 = Z-XY=0,
and one cubic surface, containing the curve, is given by
2) + PX3 + QX2Y+RXY*
Four other cubic surfaces, T71 = 0, F"2 = 0, F3 = 0, Tr4 = 0, can be obtained from t/"3 = 0 by
replacing XYby Z, respectively in, (i) the coefficient of h, (ii) the coefficient of Q, (iii) the
coefficient of /?, (iv) the coefficient of H '; these are linearly independent of I73 = 0, and of
one another. Other cubic surfaces can be obtained from U3=0 by replacing XY by Z in
two of its terms simultaneously ; for instance, if we replace XY by Z in the coefficients of
h and JT, we obtain a surface of which the equation is Vl- U3+Vt=0. Similarly all
others than #3=0, V1 = 0, ... , F4=0, are linearly deducible from these.
114. As an example of more general investigations, consider now the
correspondence between the space curve C.2p_2, for p = 4, and the original
Riemann surface. Let us seek to form a rational function having p + 1 = 5
given poles on the sextic curve. A surface of order //, can be drawn through
5 arbitrary points of the curve when //, is great enough ; we may denote
its equation by ^W=0, in accordance with § 110. It was proved that
the rational function can be written in the form c^w/vpw, <J>M being another
polynomial, of order p in the space coordinates, which vanishes in the 6/^ — 5
zeros of "^^ other than the 5 given points. Since a general surface of
order /A contains (ft + 3, 3)| terms, the most general form possible for <l>('x>,
when subject to the conditions enunciated, will contain
(At + 3, 3)-(6/x-5)
arbitrary, homogeneously entering, coefficients ; the polynomials which
multiply these coefficients, represent, equated to zero, all the linearly inde-
* § 111 preceding.
f Cf. § 108.
$ Where (/x, v) is used for the number n(/j,-l)...(/j.-v
115] PARTICULAR CURVE IN THREE DIMENSIONS. 159
pendent surfaces of order /A which vanish in the 6/u. — 5 points spoken of;
they will therefore include the
-, or *-,-,t-
surfaces of the /ith order which* contain the sextic curve. Denote the
number of these surfaces by r and their equations by U^ = 0, . . . , Ur = Q.
Then the general form of the equation of a surface, <I>W = 0, vanishing in the
6/j. — 5 given points will be
wherein \1? ..., Xr, X, /j, are arbitrary constants, and U is a surface of order /*,
other than M^', which vanishes in the 6/u. — 5 points, and does not wholly
contain the curve. The intersections of the surface ^^ =0 with the sextic
are the same as those of the surface \^M + ^U = 0 ; and the general form of
the rational function having the ^ + 1 = 5 given points as poles is
involving the right number (q + l=Q-p + l = 5-4; + l) of arbitrary
constants.
Ex. i. There are sixteen of the surfaces X*<M) + ^C/T=0 which touch the sextic (in points
other than the 6/1 - 5 fixed points).
For there are 2.5 + 2.4-2, =16, places at which the differential, dz, of the rational
function z= £7/¥M, is zero to the second order.
Ex. ii. In the example of the previous Article, prove that
86r2 Sf/3 3*73 8C72
r(*)~W''3z~W'~5z) -Asa^'
and that the integrals of the first kind, expressed in terms of X, F, Z, are given by
jfc J + X2 r+ \3Z+ X4) rfA'/A,
for arbitrary values of the constants X1} X2, X3, X4f.
115. We abstain from entering on the theory of curves in space in this
place. But some general considerations on the same elementary lines as
those referred to in §§ 81 — 83, as applicable to plane curves, may fitly
conclude the present chapter^ The general theorem considered is, that
of the intersections of a curve, in space of k dimensions, which is defined
as the complete locus satisfying k — 1 algebraic equations, with a surface
* §111.
t The canonical curve discussed by Klein, Math. Annal. t. 36, p. 24, is an immediate
generalisation of the curve Cap_a here explained. But it includes other cases also.
t See the note in Salmon, Higher Plane Curves (Dublin 1879), p. 22, "on an apparent
contradiction in the Theory of Curves" and the references there given, which include a reference
to a paper by Euler of date 1748. For further consideration of curves in space see Appendix I. to
the present volume.
160 PARTICULAR CURVE IN THREE DIMENSIONS. [115
of sufficiently high order, r, there are a certain number, P, which are deter
mined by prescribing the others, P being independent of r.
We take first the case of the curve in three dimensions, defined as the
complete intersection of two surfaces of orders m and n, say Um = 0, Un = 0.
The curve is here supposed to be of the most general kind possible, having
only such singularities as those considered in Salmon, Solid Geometry
(Dublin, 1882, p. 291). For instance the surfaces Um=Q, Un — 0 are not
supposed to touch ; for at such a place the curve would have a double point.
We prove that if r>m+n — 4, all but ^mn (m + n — 4) + 1 of the inter
sections of the curve Um = 0, Un = 0 with a surface of order r, Ur = 0, are
determined by prescribing the others, whose number is
rmn — \mn (m + n — 4) — 1.
For when, firstly, r>m + n—l, the intersections of Ur = 0 with the
curve are the same as those of a surface
Uf U in r r—in Un ' r—n U in,U n ' r — m—n = V,
wherein Vr^m, Vr_n, Vr_m_n are general polynomials whose highest aggregate
order in the coordinates is that given by their suffixes. Hence, in analogy
with the argument given in § 81, it may at first sight appear that, of the
(r + 3, 3) coefficients in Ur, we can reduce a certain number, K, given by
K= (r- m + 3, 3) + (r - n + 3, 3) + (r - m -n + 3, 3),
to zero, by using the arbitrary coefficients in Vr-m, Vr-n, Vr_m_n. This
however is not the case. For if Wr—in-n, Tr_m_n denote general polynomials,
of the orders of their suffixes, we can write the modified equation of the
surface of order r in the form
TT — TT ( V - JT W \—TT<V - 77 T }
l/r ^ m\ ' r—m u n '' r—m—n) u n \ ' r—n V m-1- r—m—n)
U m w» \ 'r—m—n " r— m— n -*- r—m—n) = "•
Now, whatever be the values assigned to the coefficients in PFr_m_n, Tr_m_n,
the coefficients in Fr_w_n— Wr-m-n— Tr_m_n are just as arbitrary as those
of Fr_m_n. And we may use the coefficients in Wr-m-n> Tr_m_n to reduce
(r — m — n + 3, 3) of the coefficients in each of the polynomials
V - JT W V - IT T
' i — m ijn''r—m—ny 'r—n '-'m-Lr—m—n)
to zero.
Hence the K equations by which we should reduce the number of
effective coefficients in Ur to (r + 3, 3) — K, are really unaltered when
2 (T — m — n + 3, 3) of the disposeable quantities entering therein, are put
equal to zero. Thus we may conclude, that so far as the intersections of Ur
with the curve are concerned, its coefficients are effectively
(r + 3, 3) - (r - m + 3, 3) - (r - n + 3, 3) + (r - m - n + 3, 3)
in number. Provided the linear equations reducing the others to zero are
116] THEOREM OF SEQUENCE. 161
independent, what we prove is that the number of effective coefficients
is certainly not more than this.
This number can immediately be seen to be equal to
rmn — \nin (m + n - 4).
Hence, we cannot arbitrarily prescribe more than rmn — \nm (m + n- 4) — 1
of the intersections of Ur = 0 with the curve.
This result is obtained on the condition that r>m + n — 1. If r = m + n— 1,
m + n— 2 or m + n — 3, the number of effective coefficients in Ur cannot
be more than in the polynomial
*J,t U m ' r—m Un V r_n,
namely, than
(r + 3, 3)-(r-m+3, 3)-(r-w+3, 3).
By the previous result this number is equal to
rmn — \mn (m + n — 4) — (r — m — n + 3, 3),
and (r - m - n + 3, 3), = (r - m - n + 1) (r - m - n + 2) (r - m - n - 3)/3 ! ,
vanishes when r = m + n—l, m + n -2, or m + w-3. Hence the result
obtained holds provided r > m + n — 4.
If we denote the number fynn (m + n — 4) + 1 by P, the result is, that
when r>m + n — 4<, we cannot prescribe more than mnr — P of the inter
sections of the curve Um = 0, Un = 0 with a surface of order r ; the prescription
of this number of independent points determines the remaining intersections.
Corollary. Hence it follows, when (?•+ 3, 3) - 1 > rmn- P + 1, that
a surface of order r described through rmn — P + 1 quite general points
of the curve, will entirely contain the curve. Hence, in general, the curve
lies upon (?- + 3, 3)-rmn+P — 1 linearly independent surfaces of order
r, r being greater than m + n — 4.
Ex. i. For the curve of intersection of two quadric surfaces, P= 1 ; every surface of
order r drawn through 4r quite arbitrary points of the curve entirely contains the curve ;
the 4r intersections of a surface of order r, which does not contain the curve, are deter
mined by 4r-l of them. When r = 2, the number (r + 3, 3)-rmn + P-I is equal to 2.
This is the number of linearly independent quadric surfaces containing the curve.
Ex. ii. For the curve of intersection of a quadric surface with a cubic surface, P = 4 ;
of the Qr intersections of the curve with a surface whose order r is >1, 6r-4 determine
the others. The number (r + 3, 3)-rwm + P-l is equal to 1 when r = 2, and equal to 5
when r=3; thus, as previously found, the curve lies on one quadric surface and on five
linearly independent cubic surfaces ; the number, for any value of r, is in agreement with
the result of § 111.
116. In regard to the intersections, with the curve, of a surface of
order m + n- 4, such a surface has effectively not more coefficients than are
contained in the polynomial
^m+n-4 U m ' n— 4 ~ U n ' m— 4>
B.
162 THERE ARE p INTEGRALS [116
for arbitrary values of the coefficients in Fn_4 and Fm_4. Here we firstly
suppose m > 3, n > 3.
Now we can prove, as before, that
(m + n-l, 3)-O-l, 3)-(m-l, 3) = %mn(m + n-4) + 1, =P.
Hence, also when m > 3 and n = 3, 2 or 1,
(TO + n _ i, 3) _ (m - I, 3), - \mn (m + n - 4) + 1 + (n - 1) (n - 2) (n - 3)/6,
is equal to P, and the number of effective coefficients in a polynomial
Um+n-4- UnVm-i, wherein the coefficients in Fm_4 are arbitrary, is as before
equal to P. Similarly for other cases.
Hence P is the number of coefficients in a polynomial Um+n-t, which are
effective so far as the intersections of the curve with the surface I7m+n_4 = 0
are concerned ; in other words, P - 1 of the intersections determine the
others. The total number of intersections is mn (m + n - 4), = 2P - 2.
The analogy of these polynomials of order m + n - 4 with the (^-poly
nomials in the case of a plane curve is obvious.
117. If now, the homogeneous coordinates of the points of the curve in
space being denoted by Xlt X2, X3, X4, the symbol [i, j] denote the Jacobian
9 ( Um, Un)/d (Xi} Xj), and (X1 + dX,, X2 + dX2, X3 + dX3, X4 + dX,) denote
a point of the curve consecutive to (Xl} X2, X3} X,), it follows from the
equations
9 Um -. v o Um , Y , ^ Um j Y — A
^+ Xs + -~ dx*~
U* 4- r 8i^
A* '
and the similar equations holding for Un, that the ratios
X2dX3- XsdX2 : X.dX.-X.dX, : X,dXz- X2dX, : X,dX.
- X.dX, : X2dX. - X<dX, : XzdX, -X.- dX3,
are the same as the ratios
[1,4]: [2, 4]: [3, 4] : [2, 3] : [3, !]:[!, 2];
each of these rows is in fact constituted by the coordinates of the tangent
line of the curve. If then u,, ua, u3, ut, v1} v.2, v3, v, denote any quantities
whatever, and, in each of these rows, we multiply the elements respectively by
U2V3 — U3V2, U3Vi — U^Va, U^ — U2V1} U^t — W4Vj, M2^4 — W4^2, ^3^4 — "5^3,
and add the results, we shall obtain for the first row
2 (u*v3 - u3v2) (X2dX3 - X3dXz) = udv - vdu,
where
u = u1Xl + u2X.> + u3X3 + u4X4, du = u^X^ + u2dX2 + u3dX3 + u4dXt, etc.,
117] WHICH ARE EVERYWHERE FINITE. 163
and, for the second row we shall obtain the determinant
t*j , U2 , U3 , *W4
o v > 3 tr . ^\r > ~^~v
O-A-i vA~2 OJL3 OJL
O "!/• i o -yr > '-\ -tr t ^V
CA. i OA. 2 0 A 3 0 A. ,
which we may denote by (uvUmUn).
From the proportionality of the elements of the two rows considered,
it follows, therefore, that the ratio (udv — vdu)/(uvUmUn) is independent of
the values of the quantities. w1} ... , v4. This ratio is of degree
in the homogeneous coordinates; namely, if Xl} X2, X3, X4 be replaced by
pXlt pX2, pX3, pXt, the ratio will be multiplied by p- <»»+»- *). Hence, if
Um+n-4 be any polynomial of degree m + n — 4, the product
Um+n-i (udv — vdu)/(uV Um Un}
is a functional differential, independent of the arbitrary factor of the homo
geneous coordinates.
The integral,
udv — vdu
ju
f TT TT \ j
(UVUmUn)
can only be infinite at the places where the curve is intersected by the
surface (uvUmUn) = 0 : if u = 0, v = 0 be regarded as the equations of planes,
this equation expresses that the straight line u = 0, v = 0, is intersected
by the tangent line of the curve at the point (Xlt X2) Xs, Z4). The
differential
udv - vdu, = 2 (uava - u3v2} (X2dX3 - X3dX2),
is zero, to the second order, when the line u=0 = v is intersected by the
tangent line, whose coordinates are X2dX3 - X3dX2> etc. Hence the ratio
(udv — vdu)/(uvUmUn) is never infinite, and the integral above is finite for all
points of the curve.
Hence*, since Um+n_^ contains P terms, we can obtain P everywhere-finite
algebraical integrals.
The same result is obtained if ult ..., v4 be polynomials in the coordinates,
MI, ..., ut being of the same degree, and v1} ... , v4 of the same degree.
As stated, we are considering a curve without singular points. If the curve had a double
point, the polynomial (uvUmUn) would vanish at that point, for all values of w1( ..., v4. We could
then prescribe Um+n_4 = Q to pass through the double point, thus obtaining a reduction of one in
the number of finite integrals. Etc.
11—2
164 TRANSFORMATION TO A RIEMANN SURFACE. [117
Ex. i. For a plane curve of order n, without multiple points, prove similarly that we
can obtain p finite algebraical integrals in the form
!>„ _ 3 (udv — vdu)l(uvf ),
where /(#!, #2, #3) = 0 is the homogeneous equation of the curve, U=u1x1+u2x2 + tiye3, etc.,
and (uvf) denotes a determinant of three rows.
Ex. ii. Shew that a surface of order m + n — 4 + fj. which vanishes in all but two of the
intersections of the curve in space with a surface of order p, £^,. = 0, is of the form
where X, X1} ... , \P are arbitrary ; and that an integral of the third kind is of the form
udv — vdu
/i
p (uvl7mUn)'
118. Retaining still the convention that u = 0, v = 0 are the equations of
planes, let u' = 0, v' = 0 be the equations of other planes whose line of inter
section does not coincide with the line u = 0 = v.
From the equations
zu-v = 0, su'-v' = 0, Um=Q, Un = 0,
wherein z, s have any values, we can eliminate the coordinates of the points
of the curve in space, and obtain a rational equation, (s, z) — 0, with which
we may associate a Riemann surface*. To any point of the curve corre
sponds a single point, z = v/u, s = v'/u, of the Riemann surface ; to any point
of the Riemann surface will in general correspond conversely only one point
of the curve in space. Hence the Riemann surface will have mn sheets,
the places, at which z has any value, being those which correspond to the
places, on the curve in space, at which the plane zu — v = 0 intersects this
curve. Thus the Riemann surface will have 2mn + 2p — 2 branch places,
p being the deficiency of the surface. These are the places where dz is zero
of the second order. Thus they correspond to the places, on the curve in
space, where udv — vdu is zero to the second order. We have seen that these
are given as the intersections of this curve with the surface (iivUmUn} = 0,
of order m + n — 2 ; their number is therefore mn (m + n — 2) = Zmn + 2P — 2.
Hence the number P, obtained for the curve in space, is equal to the
deficiency p of the Riemanu surface with which it is reversibly related.
The same result can be proved when u, v are polynomials of any, the same,
order, and u', v' are polynomials of any, the same, order.
And from the reversibility of this transformation it follows that the
everywhere-finite integrals for the Riemann surface are the same as those
here obtained for the curve in space.
* We may of course interpret the equation as that of a plane curve ; a particular case is that
in which this curve is a central projection of the space curve.
119] REFERENCES. 165
Ex. Prove that if elt e2, e3 be such that el+ez + e3=0,
(b-c)(c-a)(a-b) = (b-c)(a-d)/(e2-e3) = (c-a)(b-d)/(e3-e1) = (a-b)(c-d)!(e1-e2),
the points of the curve aX2 + bY2+cZ2+dT* = 0, X2+ Y2 + Z2+T2 = 0 can be expressed
in terms of two quantities, x, y, satisfying the equation y2 = 4 (x - ej (& - e2) (x — e3), in the
form T : X : Y : Z
=y •• V^c [(a? - etf - (e1 - ez) (et - &,)] : *J^a [(x - e2)2 - (e2 - es) (e2 - ej\
:\/a-b[(x- e3)2 - (e3 - ej (e3 - e2)].
Find x, y in terms of X, Y, Z, T in the form
(e2 - e3) Ar/ b -c + e2 (e3 - el
See Mathews, London Math. Soc. t. xix. p. 507.
119. As already remarked we have considered here only the case of a non-singular
curve in space which is completely denned as the intersection of two algebraical surfaces.
For this case the reader may consult Jacobi, Crelle, t. 15 (1836), p. 298 ; Pliicker, Crelle.
t. 16, p. 47 ; Clebsch, Crelle, t. 63, p. 229 ; Clebsch, Crelle, t. 64, p. 43 ; Salmon, Solid
Geometry (Dublin, 1882), p. 308 ; White, Math. Annal. t. 36, p. 597 ; Cayley, Collected
Works, passim. For the more general case, in connexion however with an extension of the
theory of this volume to the case of two independent variables, the following, inter alia,
may be consulted : Noether, Math. Annal. t. 8 (1873), p. 510 ; Clebsch, Comptes Rendus de
I'Acad. des Sciences, t. 67, July— December, 1868, p. 1238 ; Noether, Math. Annal. t. 2,
p. 293, and t. 29, p. 339 (1887) ; Valentiner, Acta Math. t. ii. p. 136 (1883) ; Halphen,
Journal de VEcole Polyt. t. lii. (1882), p. 1 ; Noether, Abh. der Akad. zu Berlin (1882) ;
Cayley, Collected Works, Vol. v. p. 613, etc. ; and Picard, Liouv. Journ. de Math.
1885, 1886 and 1889.
Ex. i. Prove that
(r+t, k)-2(r + k-
where (r, M) denotes r(r- !)...(/•- /u+ 1)/M !, m1( ... , mk_^ k are any positive integei-s, r is a
positive integer greater than mx+ »i2+... +mk_1- k - 1, 2 denotes a summation extending
to all the values i=l, 2, ..., (k- 1), 2 denotes a summation extending to every pair of two
2
unequal numbers chosen from the series m^ m2, ..., mk_l, and so on. Hence infer that
of the intersections of a general curve in space of k dimensions, which is determined as the
complete locus common to k-l algebraic surfaces of orders mlt m2, ..., mk_l, with a
surface of order r, all but
are determined by the others. The result is known to hold for £ = 2. We have here been
considering the case £ = 3.
Ex. ii. With the notation and hypotheses employed in Salmon's Solid Geometry (1882),
Chap. XII. (p. 291) (see also a note by Cayley, Quarterly Journal, t. vn., or Collected Works,
Vol. v. p. 517), where m is the degree of a curve in space, n is its class, namely the number
of its osculating planes which pass through an arbitrary point, r is its rank, namely the
number of its tangents which intersect an arbitrary line, a is the number of osculating
planes containing four consecutive points of the curve, /3 the number of points through
which four consecutive planes pass, x the number of points of intersections of non-consecu-
166 EXAMPLES. [119
tive tangents which lie in an arbitrary plane, y the number of planes containing two non-
consecutive tangents which pass through an arbitrary point, h the number of chords of the
curve which can be drawn through an arbitrary point, g the number of lines of intersection
of two non-consecutive osculating planes which lie in an arbitrary plane, ^ the number of
tangent lines of the curve which contain three consecutive points, prove, by using Pliicker's
equations (Salmon, Higher Plane Curves, 1879, p. 65) for the plane curve traced on any
plane by the intersections, with this plane, of the tangent lines of the curve in space, that
the equations hold,
(1) n = r(r-l)-2x-3m-3^, (3) r = n(n- 1) -2#-3a,
(2) a = 3r(r-2)-6^-8(m+^), (4) m+^ = 3n(n-2)-Gg-8a,
pl-l=%r(r-2>)-x-m — ^=%n(n — S)—g — a ..................... (A),
p1 being the deficiency of this plane curve.
Prove further, by projecting the curve in space from an arbitrary point, and using
Pliicker's equations for the plane curve in which the cone of projection is cut by an
arbitrary plane, the equations
(5) r = m(m-I)-2h-3p, (7) m = r (r- l)-2#-
(6) S.+?i = 3m(m-2)-6A-8/3, (8) /3 =
/>2-l = im(»i-3)-A-0 = ir(r-3)-y-n-$. .................... (B),
p2 being the deficiency of this plane curve.
From the equations (1) and (7) we can infer n - m = 3n — 3m — 2 (x-y\ and therefore
Hence Pi=pz.
Ex. iii. For the non-singular curve which is the complete intersection of two algebraic
surfaces of orders p, v, prove (cf. Salmon, Solid Geometry, pp. 308, 309) that in the notation
of Ex. ii. here,
Hence, by the equations (B) of Ex. ii. prove that, now,
p^Pz^pvfa + v-ty + l.
This is the number we have denoted by P.
Ex. iv. Denoting the number pl=p2j in Ex. ii., by p, prove from equations (5) and (B)
that
= 3 (r+j8-2m).
Hence shew that if, through a curve C of order m, lying on a surface S of order /n, we
draw a surface of order v, cutting the surface S again in a curve C" of order m', and if
p, p' denote the values of p for these curves C, C' respectively, then
(see Salmon, pp. 311, 312). Shew that each of these numbers is equal to the number, i,
of points in which the curves C, C' intersect, and interpret geometrically the relation
i+r + 0=m(fji + v-2').
Ex. v. If in Ex. iv. a surface <£ of order p + v - 4 be drawn through (/* + v — 4) m' —p' + 1,
or i—l+p', of the points of the curve C", prove that, so far as its intersections with the
curve C are concerned, the surface (f) contains effectively p terms. Prove further that 0
contains the curve C' entirely.
119] EXAMPLES. 167
Ex. vi. Prove that a surface of order p + v — 4 passing through i— 1 of the intersections
of the curves 0, C', in Ex. iv., will pass through the other intersection.
Ex. vii. An example of the case in Ex. iv. is that in which /i = 2, i/ = 2, m = 3, m' = l.
Then C" is a straight line and p' = 0 : hence p is given by — 2 = 2p - 2. Hence, for the
cubic curve of intersection of two quadrics having a common generator, p = Q. And in fact
coordinate planes can be chosen so that the homogeneous coordinates of the points of the
cubic can be expressed in the form
X : Y : Z : T=\ : 6 : 0* : <93,
6 being a variable parameter. For instance (using Cartesian coordinates) the polar planes
of a fixed point (X'Y'Z1) in regard to quadrics confocal with X2/a + Yzjb + Z*/c = l are the
osculating planes of such a cubic curve, the coordinates of whose points are expressible in
the form
XX' = (a + \}*l(a-b)(a-c\ YY' = (b + \)3l(b-c)(b-a), ZZ' = (c + \)3/(c-a)(c-b),
\ being a variable parameter.
Ex. viii. For the quintic curve of intersection of a quadric and a cubic surface having
a common generator we obtain, from Ex. iv., putting m'=l, p' = 0, m = 5, that p = 2 ; the
results of Exx. iv., v., vi. can be immediately verified for this curve ; further, if the surfaces
be taken to be yU-zV=0, yS-zT=0, where U, V are of the first degree in x, y, z and
S, T of the second degree, and we put y = zg, x=zrj, we obtain
where the Greek letters au a2, . . . denote polynomials in £ of the degrees of their suffixes.
Hence, if a- be defined by the equation,
XI(T = 2? (X^ft +X1a1y1 + Vi2) + Vfo + Xi (al72 + «27i) + 2Vi«2 ,
we obtain o-2 = (£, 1)6 ; £, a- are rational functions of x, y, z and x, y, z are rational functions
of £, o-.
Ex. ix. Prove that if the sextic intersection of a cubic surface and a quadric surface,
break up into a quartic curve and a curve of the second order, the numbers p, p' for these
curves are p = l, p' = Q or p=0, p'= - 1 according as the curve of the second order is a
plane curve or is two non-intersecting straight lines.
Ex. x. In analogy with Ex. iv., shew that the deficiencies of two non-singular plane
curves of orders m, m' are connected by the equation
m (m + m' - 3) - (2jo - 2) = mm' = m' (m + m' - 3) - (2p' - 2),
and further in analogy with Ex. v. that if a plane curve, of order m+m' -3, be drawn
through (m+.m1 - 3) m' -p'+l independent points of the curve of order m', only p - I of its
intersections with the curve of order m can be prescribed.
Further indications of the connexion of the theory of curves in space with the subject
of this chapter will be found in Appendix I.
[120
CHAPTER VII.
COORDINATION OF SIMPLE ELEMENTS. TRANSCENDENTAL UNIFORM
FUNCTIONS.
120. WE have shewn in Chapter II. (§§ 18, 19, 20), that all the funda
mental functions are obtainable from the normal elementary integral of the
third kind. The actual expression of this integral for any given form of
fundamental equation, is of course impracticable without precise conventions
as to the form of the period loops, and for numerical results it may be more
convenient to use an integral which is denned algebraically. Of such
integrals we have given two forms, one expressed by the fundamental
integral functions (Chap. IV. §§ 45, 46), the other expressed in the terms of
the theory of plane curves (Chap. VI. § 92, Ex. ix.). In the present Chapter
we shew how from the integral P^'", obtained in Chap. IV.*, to determine
algebraically an integral Qf' a for which the equation Qx' a = Qz'c has place ;
incidentally the character of P*' " , as a function of z, becomes plain ; and
therefore also the character of the integral of the second kind, £%' a , which
was found in Chap. IV. (§§ 45, 47).
This determination arises in close connexion with the investigation of
the algebraic expression of the rational function of x which was obtained in
§ 49 and denoted by -fy (x, a ; z, clt ... cp). It was there shewn that every
rational function of x can be expressed in terms of this function. It is shewn
in this Chapter that any uniform function whatever, which has a finite
number of distinct infinities, which may be essential singularities, can be
expressed by such a function.
Further, it is here shewn how to obtain an uniform function of x having
only one zero, at which it vanishes to the first order, and one infinity ; and
that any uniform function can be expressed in factors by means of this
function.
* For the integral of the third kind obtained in Chap. VI. the reader may compare Clebsch
and Gordan, Theorie der Abel. Functionen (Leipzig, 1866), p. 117, and, for other important results,
Noether, Math. Annul, xxxvn. (1890), pp. 442, 448; also Cayley, Amer. Journal, v. (1882), p. 173.
122] NOTATION. 169
121. Let Wi'a, ...,Upa denote any p linearly independent integrals of
the first kind, vanishing at the arbitrary place a. Let t denote the infinit
esimal at x, and let Du*, ...... , Du* denote the differential coefficients of the
integrals in 'regard to t, all of which are everywhere finite. Let d , ..., cp
denote any p fixed places of the Riemann surface, so chosen that no linear
aggregate of the form
where Xx, ..., \ are constants, vanishes in all the places c,, ..., cp, but such
that one linear aggregate of this form vanishes in every set of p — 1 of these
places*; and let Wi(x) denote the linear aggregate, of this form, which
vanishes in all of c1} ..., cp except d, and is equal to 1 at the place c;.
Then Wi(x) is expressible as the quotient of two determinants; the
denominator has Dusr for its (r, s)th element, the numerator differs from the
denominator only in the t'-th row, which consists of the quantities Du[ , . . . ,
DM* ; thus w1(x), ..., (Op(x) are determinable algebraically when «*, ... , uxp are
given. Conversely the differential coefficients of the normal integrals of the
first kind (§§ 18, 23) are clearly expressible by wl(x), ..., wp(x), in the form
H; O) = wj (x} £li (d) + ...... + cop (x) fli (Cp).
We have already used ef* as a notation for the normal integral
i rx fx
H- . I nt (ai) dtx. In this chapter we shall use the notation V*'a = I ca{ (x) dtx.
67TI J a J a
If the period of the integral u^ at the j-ih period loop of the first kindf
be denoted by Citj, we can express vfa as the quotient of two determinants,
the denominator having Cjti for its (i,j)tfi element, and the numerator being
different from the denominator only in the tth row which consists of the
elements u*'a, ..., ux' a.
122. Consider now the function of # expressed j by
p
rx, a -s? /_\ -p-r, a
' - Z a)r(z)L ,
r=\
z being any place whatever. The function is clearly infinite to the first
order at the place z, like —t~l, tz being the infinitesimal at z ; it is also
infinite at each of the places cl} ..., cp, and, at Cj, like &i(*)£ , tc. being the
infinitesimal at C;. The function has no periods at the period loops of the
* Thus there exists no rational function infinite only to the first order at each of clt ..., cp.
Cf. §§ 23, 26.
t C(i> is the quantity by which the value of u*'a on the left side of this period loop exceeds
the value on the right side. See the figure, § 18, Chap. II.
t Klein, Math. Annul, xxxvi. p. 9 (1890), Neumann, loc. cit. p. 14, p. 259.
170 A FUNDAMENTAL FUNCTION [122
first kind. At the tth period loop of the second kind the function has the
period
fM*)- 1 r^ooao,.),
r=\
which, as remarked (§ 121), is also zero. Hence the function is a rational
function of x. It vanishes at the place a. We shall denote the function by
•fy (x, a; z, c1} ..., Cp). It is easy to see that it entirely agrees, in character,
with the function given in § 49.
For the places c1; ..., cp have been chosen so that no aggregate of the
form
Xjflj (#) + + XpOj, (x)
vanishes in all of them. Hence (Chap. III. § 37) the general rational function
having poles of the first order at the places z, c1} ... , cp is of the form Ag -f B,
where g is such a function, and A, B are constants. These constants can be
uniquely determined so that the residue at the pole, z, is — 1, and so that
the function vanishes at the place a.
Ex. For the case^> = l, if we use Weierstrass's elliptic functions, the places x, a, z, c,
being represented by the arguments u, a, v, y1} and put x = $u, y = %>'(u) etc., we may
take, supposing v not to be a half period,
and obtain
or
Vr(#, a; z, Cj) =
and any doubly periodic function can be expressed linearly by functions of this form,
in which the same value occurs for yx and different values for v. (Of. § 49, Chap. IV.)
123. Since mdz), = ^r Vs' c, is a linear function of ^(z), ..., Qp(z), it
atz *
follows that &>;(»/j7 is a rational function of z\ and r*'a, = -^- H^'*,
I at °*s
= (ux,*\< is such that* Tx'a 1- is a rational function of z\ hence
\dz *> c J dt, " I dt
* Throughout this chapter such an expression as f(z) — is used to denote the limit, when a
variable place £ approaches the place z, of the expression /(£) ^-, t being the infinitesimal for
124] WITH p + 1 POLES. 171
Idz
i/r (x, a; z, d, ...,Cp)j-jr is a rational function of z. It is easy also to see,
dc-
from the determinant expression of W{ (z), that &>j (z) -~ is a rational function
cut
of cn ..., cp.
Hence i/r (x, a; z, d, ..., cp)/ -j- is a rational function of the variables of
all the places x, a, z, d , . . . , cp.
Further, as depending upon z, i/r (x, a ; z, c1} ..., cp) is infinite only when
F*' a is infinite ; and F*' a, = -j- Uz'a, is infinite only when z is at x or at a.
atz x> a
At the place x, Tx'a is infinite like -7- log tx, namely like the inverse of the
atx
infinitesimal at the place x.
Hence ty (x, a ; z, cl , . . . , cp), regarded as depending upon z, is infinite only
when z is in the neighbourhood of the place x, or in the neighbourhood of the
place a. At the place x, ^r(x, a; z, c1; ..., cp) is infinite like the positive
inverse of the infinitesimal, at the place a it is infinite like the negative inverse
of the infinitesimal. The rational function of z denoted by
i / \ Idz
yfca; z, d, ..., Cp)l fa
will therefore be infinite at the place x like T and at the place a
wl + 1 z — x
like , where w^ + 1, w.2 + 1 denote the number of sheets that
w2 + L z — a
wind at the places x, a respectively; and will be infinite at every branch
A
place, like , ^-w, t being the infinitesimal at the place, w + l the number
\U) -7- 1 j t
of sheets that wind there, and A the value of ty(x, a; z, cl} ..., cp) when z is
at the branch place.
The actual expression of the function i/r (x, a ; z,cly ..., cp) is given below
(S 130).
124. From the function ty(x, a ; z, cl} ..., cp) we obtain a function,
„ t N J**(jf' a; *• °" - • c"> dt* n*f. a - I v'r- c r*' a
E(x,z) = e ,=e* r=i
wherein c is an arbitrary place, which has the following properties, as a
function of x.
the neighbourhood of the place z. When z is not a branch place - = 1 ; when w + l sheets wind
ttt
at Z> dt = ^W + 1^" ^cf' ^ 2' 3; CliaP- I-)- AmPle practice in the notation is furnished by the
examples of this chapter.
172 UNIFORM FUNCTION WITH ONE ZERO. [124
(i) It is an uniform function of x. For the exponent has no periods at
the period loops of the first kind, and at the iih period loop of the second
kind it has the period
27T^'C- I V^n^Cr)
r=l
which, as follows from the equation
n,; (Z) = «! (Z) flf (Cl) + ...... + 0)p(z)fli (Cp),
is equal to zero. Further the integral multiples of 27rt, which may accrue
to Hx' a when x describes a contour enclosing one of the places z, c, do not
Z, C
alter the value of the function.
(ii) The function vanishes only at the place z, and to the first order.
(iii) The function has a pole of the first order at the place c.
(iv) The function is infinite at the place d, like evi tci , tc. being the
infinitesimal at the place. We may therefore speak of c1} ..., cp as essential
singularities of the function.
125. In order to call attention to the importance of such a function
as this, we give an application. Let R (x} denote a rational function, having
simple poles at a.l, ..., ctm, and simple zeros at /?1; ..., /3W. We suppose these
places different from the fixed places c, a, c1} ..., cp. Then the product
is an uniform function of as, which becomes infinite only at the places c1} ... cp ;
at Ci it is infinite like a constant multiple of
Now, in fact, \ogF(x) is also an uniform function of x: for it is only
infinite at the places clf ..., cp, and, at the place d, like — ( X V°?' r\ F^a.
\r=l '
r rpf (x\
Hence the integral U log F(as\ = -TTT^/ dx, taken round any closed area
J J F(x)
on the Riemann surface which does not enclose any of the places Ci, . .., cp, is
m a f (It
certainly zero, and taken round the place d is equal to — 2 V ^ r I ~ _', taken
r=l J t
Ci
round d, and is, therefore, also zero.
But an uniform function of x which is infinite only to the first order at
each of cls ..., cp does not exist. For the places c1} ..., cp were chosen
so that the conditions that the periods of a function, of the form
125J FACTORIZATION OF ANY RATIONAL FUNCTION. 173
wherein \lt ... ,\p are constants, should be zero, namely the conditions
\A.(d)+ ...... + V^- (CP) = °> r=l, 2, ...... ,p,
are impossible unless each of Xn . .., \p be zero.
m
Hence we can infer that S Va.r> r = 0, for i = I, 2, . . . , p, and that F (x) is
r=l
a constant; this constant is clearly equal to F (a), for E (a, z) — 1 for all
values of 2.
Hence, any rational function can be expressed as a product of uniform
functions of x, in the form
where a1, ...,ocm are the poles and /31; ..., ftm the zeros of the function. We
have given the proof in the case in which the poles and zeros are of the first
order. But this is clearly not important.
Further, the zeros and poles of a rational function are such that
2 V"C = 2 V?r'°, i=l, 2, ...,p,
r=l r=l
c being an arbitrary place. This is a case of Abel's Theorem, which is to be considered in
the next Chapter. We remark that in the definition of the function JS(x, z) by means of
Eiemann integrals, the ordinary conventions as to the paths joining the lower and upper
limits of the integrals are to be regarded ; these paths must not intersect the period loops.
r, • r, , , „:*;, a , (X — Z a — C\ . -p. . (x — Z) (« — C]
Ex. i. For the case « = 0, n =log ( and E (x. z)=± J-± '
z,c ° \x-ca-zj (x-c)(a-z)
Ex. ii. For the case p — l, supposing the place c represented by the argument y, we
have
fz (v
= t(*t«: *»ci» ...,<V)&-»-
J c J y
and therefore
a- (u — y) a- (a — v)
Ex. iii. Prove, if a', cf denote any places whatever, that
E(x,c'}E(a',z)~
Ex. iv. The rational function of #, ^(x, f ; z, clt ... , cp), will, beside f, have p zeros,
^y Yi> ••• » 7p> such that the set f, yu ... , yp is equivalent with or coresidual with the set
z, c,, ... , cp (§§ 94, 96, Chap. VI.). Hence, in the product
^ (x> f > zi c\-> ••• > CP) "A (-r> z\ C> 7i> ••• > VP)>
174 MODIFICATION WHEN THE FIXED PLACES [125
the zeros of either factor are the poles of the other, and the product is therefore a constant.
To find the value of this constant, let x approach to the place z. Then the product
becomes equal to
- tx-1 . tx [Dx^ (x,z; £ 7l, ... , yv}]x=z .
It is clear from the expression of >//•(#, a ; z, clt ... , cp) which has been given, that
Dx^f(x, a ; z, ct, ..., cp) does not depend upon the place a. Thus, by the symmetry, we
have the result
z; f, yi, ..., yp) = - D,+ (z, a ; £ yn ..., yp)
where a is a perfectly arbitrary place, and the sets z, clt ..., cln f, yu ..., yp are subject to
the condition of being coresidual.
Hence also if W(x ; z, c1, ..., cp) denote the expression
Dx [+ (x, a; z, c1? ... , cp) - 1^1 a] ,
we have
W(z; f, yj, ..., yp) = W(£ ; «, c^ ..., cp),
provided only the set 2, clt ..., cp be coresidual with the set f, yn ..., yp.
Ex. v. Prove, with the notation of Ex. iv., that
^(x, a; z, clt ..., cp)^(a, a; f, yl5 ..., yp) = ^(^, f; 2,^, ...,cp)^(^, a; £ y1? ...,yp).
126. These investigations can be usefully modified*; we can obtain
a rational function ty (x, a ; z, c), having the same general character as
•fy (x, a; z, Cj, ..., Cp) but simpler in that its poles occur only at two distinct
places z, c, of the Riemanu surface, and we can obtain an uniform function
E (x, z) having only one zero, of the first order, at the place z, which is
infinite at only one place, c, of the surface.
The limit, when the place x approaches the place c, of the rth differential
coefficient of £li(x) in regard to the infinitesimal at the place c, will be
denoted by ftfCc), or simply by n<r). We have shewn (Chap. III. § 28)
that there are certain numbers kL> ..., kp, such that no rational function
exists, infinite only at the place c, to the orders k1} ..., kp. The periods of a
function of the form
T\k - 1 -pia;, a ^ T^fc, - 1 r\x, a •* Tjfc/j-1 -pa;, a
Vc lc ~^L'c Lc ~ ...... -^P^e Lc '
wherein Xlt ...,\p are constants, and 2)kc~lTxc'a denotes f the limit, when z
approaches c, of the &th differential coefficient of the function IF' " in regard
to the infinitesimal at c, /u, being an arbitrary place, are all of the form
These periods cannot all vanish when k is any one of the numbers
klt ...,kp; thus the determinant formed with the p2 quantities OJ does
* Giinther, Crelle, cix. p. 199 (1892).
t For purposes of calculation, when c is a branch place, it is necessary to have care as to the
definition.
127] COALESCE AT ONE PLACE. 175
not vanish ; but \ , . . . , \p can be chosen to make all these periods vanish
when k is not one of the numbers klt . . . , kp.
127. Consider now the function
i / \ T&> a o / ,v\ r» / ~\
•yr(x,a; z,c)= L , ili(z) , • , *«yvf]
,(*r-D
wherein r = 1, 2, . . . , £>.
Since the period of Tx' a, at the ith period loop of the second kind, is
£li(z), the periods of the elements of the first column of the first deter
minant are the elements of the various other columns of that determinant.
Thus the function is a rational function of x.
We shall denote the minors of the elements of the first column of the
first determinant, divided by the second determinant, by 1, — twj (2), ..., — a)p (z),
although that notation has already (| 121) been used in a different sense.
Before, o.\ (z) was such that o>; (cr) = 0 unless r = i in which case Wi (d) = 1 ;
now, as is easy to see, [-D*7""1 Wi (z)~]g=e is 0 or 1 according as r is not equal or
ft
is equal to i. The integrals coi(z)dtz are linearly independent integrals of
the first kind (cf. Chap. III. § 36).
Then the function can be written
t=i
~l
the function is infinite at z like — t~l, tz being the infinitesimal at the
place z, and is infinite at c like *
tc being the infinitesimal at the place c. It is not elsewhere infinite. The
function vanishes when as approaches the place a. As before (§ 123)
fdz
•fy(x,a; z, c)/-jr is a rational function of all the quantities involved; and
/ CLt
•^ (x, a ; z, c), as depending upon z, is infinite only at the places x, a, in each
case to the first order.
* This is clear when c is not a branch place, since then, when x is near to c, r*'" is infinite
like -- — ; and the (fc-l)th differential coefficient of this in regard to c is - \k-l(x-c)~*.
When c is a branch place, exactly similar reasoning applies if we first make a conformal repre
sentation of the neighbourhood of the place, as explained in Chap. II. §§ 16, 19.
176 MODIFIED FUNCTION WITH ONE ZERO. [128
128. If now R(x) be a rational function with poles of the first order at
the places zlt ..., zm, it is possible to choose the constants \, ..., \p so that
the difference
R (as) - Xi^/r (x, a ; zly c)-\^(x, a; z2,c)- ...... -Xm>/r(#, a; zm, c)
is not infinite at any of the places zlt ..., zm; this difference is therefore
infinite only at the place c, and is infinite at c like
-(A, l&j-l*;*1 + ...... + Ap 'fep-l t;kp),
where
At = X1&>; (^) + ...... + \mCOi (>m), (i = 1, 2, . . . , p).
But, a rational function whose only infinity is that given by this ex
pression, can be taken to have a form
wherein A is a constant; and we have already remarked (§ 126) that the
periods of this function cannot all be zero unless each of Alt ..., Ap be zero.
Hence this is the case, and we have the equation
R(x) = A + \-^(x, a; zlt c) + ...... + \mty(x, a; zm, c),
whereby any rational function with poles of the first order is expressed by
means of the function ty(x, a; z, c). It is immediately seen that the
equations Al = Q=... = Ap enable us to reduce the constants X^ . .., Xm to
the number given by the Riemann-Roch Theorem (Chap. III. § 37).
When some of the poles of the function R (x) are multiple, the necessary
modification consists in the introduction of the functions
Dzty (x, a ; z, c\ D\ty (as, a; z, c), .......
Ex. If »!(#), •••>«*(#) denote what are called <at(o;), ..., u»p(x) in § 121, and the
notation of § 127 be preserved, prove that
P ki~l_
mr(z)= 2 u>i(z)Dc u>r(c),
t=l
and that
129. From the function ^ (x, a ; z, c) we derive a function of ar, given by
fV(ar,a;«,c)««» Hx-a - 2 F2'cD*r"'r^"
E (x, z) = e ° , = e *' »-=i
rg
where, in the notation of 8 127, V*'e = a)r(z}dtz, which has the following
-'c
properties :
(i) It is an uniform function of x ; there exists in fact an equation
• z,c •£> T7*'
iv, =» 2« r_
r=l
130] ALGEBRAICAL EXPRESSION OF THE FUNCTIONS. 177
(ii) The function vanishes to the first order when the place x approaches
the place z ; and is equal to unity at the place a.
(iii) The function is infinite only at the place c, and there like
_! I Vz;°\kr -It;*-
tc er=l
As before we can shew that any rational function R(x), with poles at aly ..., a,n, and
zeros at fa, ..., /3m, can be written in the form
this being still true when some of the places au ...,<%, or some of the places fa, ..., /3OT
are coincident.
130. We pass now to the algebraical expression of the functions which
have been described here*. We have already (Chap. IV. § 49) given the
expression of the function \|r (x, a; z, c1} ..., cp) in the case when all the
places a, z, cl5 ..., cp are ordinary finite places. In what follows we shall
still suppose these places to be finite places; the necessary modifications
when this is not so can be immediately obtained by a transformation of
the form x = (£ — k)~l, or by the use of homogeneous variables (cf. § 46,
Chap. IV., § 85, Chap. VI).
If, s being the value of y when x = z, we denote the expression
byf (z, x), and use the integrands &>i(#), ..., Q>p(x) defined in § 121, the
rational expression of ty(x, a; z, C1} ..., cp), which was given in § 49, can be
put into the form
g
^ (x, a ; z, d, . . . , Cp) = (z, x) - (z, a) - z wr (z) [(ci} x) - (a, a)].
r=\
In case z be a branch place, the expression (z, x) is identically infinite in
virtue of the factor f (s) in the denominator, and this expression can no
longer be valid. But, then, the limit, as £ approaches z, of the expression
* It is known (Klein, Math. Annal. xxxvi. p. 9 (1890); Giinther, Crelle,cix. p. 199 (1892)) that
the actual expressions of functions having the character of the functions i//(x, a; z, clt ..., cp),
E (x, z), Qr i have been given by Weierstrass, in lectures. Unfortunately these expressions have
not yet (August, 1895) been published, so far as the writer is aware. Indications of some value
are given by Hettner, Gotting. Nachr. 1880, p. 386; Bolza, Gotting. Nachr. 1894, p. 268;
Weierstrass, Gesamm. Werke, Bd. ii. p. 235 (1895), and in the Jahresbericht der Deuts. Math.-
Vereinigung, Bd. iii. (Nov. 1894), pp. 403—436. But it does not appear how far the last of these
is to be regarded as authoritative ; and it has not been used here. The reader is recommended
to consult the later volumes of Weierstrass's works.
t This notation has already been used (§ 45). It will be adhered to.
B. 12
178 ALGEBRAICAL EXPRESSION OF THE FUNCTIONS. [130
jit
(£, x) -~ , wherein t is the infinitesimal at the place z, is finite* ; if we denote
(MI
dz
this limit by (z, x) -j- , and introduce a similar notation for the places
uit
Cj, . .., cp, we obtain the expression
i/r 0, a ; z, d, . . . , Cp) = [(z, x) - (z, a)] ^ - 2 a)r (z} . [(a, x) - (a, a)] ~ ,
which, as in § 49, has the necessary behaviour, for all finite positions of
z, d, GI , . . . , Cp.
From this expression we immediately obtain (§ 45)
131. In a precisely similar way it can be seen (see § 127) that
^ (x, a ; z, c) = [(z, x) - (z, a)] ^ - 2 <*>r (z) D*J~l j[(c, as) - (c, a)] ^4 ,
wherein DkJ~l |[(c, x) - (c, a)] ^| = limit^ [(^J^ {[(£ *) ~ (S «)] §}] 5
for this expression can be written as the quotient of two determinants, in
the manner of § 49, and the integrands f^ (z), . . . , £lp(z) are linear functions
of the p integrands
<£j (z) dz z<f>-i (z) dz z*1*1 <f>! (z) dz </>2 (z) dz
/'(«)#' T7!? dt' "" f («) dt' f(s) di' '
these latter quantities can therefore be introduced in the determinants in
place of Hj (z), . . . , £lp (z), the same change being made, at the same time,
for the quantities ^(c), ..., Op(c), throughout. Then it can be shewn
precisely as in § 49 that the expression is not infinite when x is at infinity.
In regard to finite places, it is clear that the expression
Tfr-1 it(r <r\-(r a\\ — I - 7)** P°'y
U9 \ [\c, x) \c, a)] , f, - uc jrXt a ,
regarded as a function of x, has the same character, when x is near to c, as
the function Dfcr~1ra;'a.
c c
Hence, also, it follows that E(x, z) has the form
* /' M, when r) is very nearly s, vanishes to order i + w, and dfldt to order w (see Chap. VI.
§ 87). Or the result may be seen from the formula
(Chap. IV. § 45).
132] EXAMPLES.
132. Ex. i. For the case (p = 1) where the surface is associated with the equation
f = (x, 1)4,
if the values of the variables x, y at the place a be respectively a, b, and the values at the
place q be ct, dv respectively, then
d, dz . . s+y
(a) when (cls d\) is not a branch place CDI(S) = - ^ , ft, ^) = 2s(2_
and
r s+y s + b__~\dz _djdzr _d1+y__ dt + b
('r' a '• *> ci) = \Zs(z-x} ~ 2s (z-a)] dt s ~dt [.2^ (c, - x) 2^ (ct - a) J
_ J. dz rs+y _ s + b _ d^+y d±
~
z-a
(/3) when (ct, d\) is a branch place, in the neighbourhood of which
M-- — C a?^ = limit of — ^^— .2*= — ^—
and
•fy (so, a ;
_?±b_~\(k_^(!lz(__y__ b }
2* (z - a) J dt 2s dt \A (q - x} A (^ - a) j
_ _! dz (s+y _ s + b _ y b |
~ 2s dt [z - x z-a c-^ — x c^ — a)
If (s, 2) be not a branch place, ^ jf = ^ ; if (5, 2) be a branch place, in the neighbour-
1 dz .. ., ,, 1 ,,. 1
hood of which x=z+t\ y = Bt + ... , ^ ^ , = limit of ^ 2«, =-g.
Ex. ii. For the case (/> = 2) where the surface is associated with the equation y2 =/(#),
where f(x} is an integral function of x of the sixth order, we shall form the function
^(x, a; z, cn 02) for the case where clt c2 are branch places, so that /(c1)=/(c.2) = 0, and
shall form the function \ff(x,a; z, c) for the case when c is a branch place, so that/(c)=0.
When c1; c2 are branch places, in the neighbourhood of which, respectively, 0;=^+^,
y = Altl + ..., andtf=c2+«22, y = Afa + ..., so that Al2=f'(c1~), A<?=f'(c2), we have
and
r s+y s + b "1 dz _ l_ dz /_g^c2 ( J/_ b \
^a;*C|>^"ia«(«-*) 2s(2-«)J^ sUfftK-^V^-* q-«/
When c is a branch place, in the neighbourhood of which .r=<
so that A*=f (c), the numbers klt k2 are 1, 3 respectively (Chap. V. § 58, Ex. ii.). In the
definition of the forms ^(z), o>2(2) (§ 127) we may, by linear transformation of the 2nd,
3rd, ..., (/) + l)th columns of the numerator determinant, and the same linear transforma
tion of the columns of the denominator determinant, replace Ql(z), ..., Qp(z) by the
differential coefficients of any linearly independent integrals of the first kind. In the case
now under consideration we may replace them by the differential coefficients of the
integrals I — , 1-5- . Hence the denominator determinant becomes
12—2
180
limit, =e
HYPERELLIPTIC CASE.
[132
dt
x dx = limits =c
2w dt
' \dt
c5\
4J!
Hence
= limit,. =
•\_dz
2s dt
z dz
.
'"A*
and
1 dz z dz
2s dt' 2* dt
I dx xdx
1 dzc — z
2s eft ^T
Hence
1 dz
Further
but
Hence the function ^ (a?, a ; s, c) is given by the expression
s+b
_2«(f -*) 2s (2
b ~]dz -
- a) J dt
A 2s dt \c-x
z — c
1 dz(A-B(x-c) A-B(a-c} \
2s ^\ (^--c)2 y (a-c)2 /'
JKr. iii. Apart from the algebraical determination of the function ^(x, a; z, cls ..., cp)
which is here explained, it will in many cases be very easy* to determine the function by
the methods of Chapter VI. It is therefore of interest to remark that, when the function
VT (a?, a ; z, clf ... , ep) is once obtained the forms of independent integrals of the first and
second kinds can be immediately obtained as the coefficients in the first few terms of the
expansion of the function in the neighbourhood of its poles, in terms of the infinitesimals
at these poles.
* An adjoint polynomial * of grade (n-l)<r + n-2 which vanishes in the p + 1 places
z, clt .... cp will vanish in n+p-3 other places. The general adjoint polynomial of grade
(n-1) ff + n-2 which vanishes in these n+p-3 places will be of the form \* + /*e, where X and
/t are constants. The function f(x, a ; z, clt .... c,) is obtained from X + /t0/*, by determining
X and yu properly. Cf. Noether (toe. cit.) Math. Annal. xxxvii,
132] THE FUNCTION i|r IS FUNDAMENTAL. 181
In fact, if tj, be the infinitesimal in the neighbourhood of the place ci} arid Mr, ; denote
Ci, x)
D<\_(C" 'i)
the expansion of >/<•(#, a; z, c1? ..., cp), as a function of x, in the neighbourhood of the
place Ci, has, as the coefficient of ti~l, the expression o>i (z), which is one of a set of linearly
independent integrands of the first kind, while the coefficient of ^ is
Now the elementary integral of the second kind obtained in Chap. IV. (§§ 45, 47)
with its pole at a place c, when z is the current place, is E^ a = I dzDc(z, c), whether c be
J a
a branch place or not, and when z is near a branch place this must be taken in the form
Hence the coefficient of ^ in the expansion of ^(x, a; z, cl} ..., c^,), when x is near to cit
is equal to
DtEla - S «,(*)*•„„
* r=\
This is the differential coefficient of an integral of the second kind, with its pole at cf ,
the current place being z. We shall see that the integral of the second kind with its pole
at any place 2 can be expressed by means of the functions Ee , ..., Ee (§135, Equation x.).
Ex. iv. Similar results hold for the expansion of the function ^ (x, a ; z, c), as a func
tion of x, when x is in the neighbourhood of the place c. If tc be the infinitesimal at this
place, the terms involving negative powers are
of which the coefficients of the various powers of tc are differential coefficients of linearly
independent integrals of the first kind ; the terms involving positive powers are
where Piy k is the limit, when the place .v approaches the place c of the expression
Among the coefficients of these positive powers of te, only those are important for
which t is one of the numbers ^ , . . .,{;„. This follows from the fact that Dek ~ l r*' a, when
k is not one of the numbers *lf ..., kp, is expressible by those of
of which the indices ^- 1, kz- 1, ..., are less than k- 1, together with a rational function
of x (Chap. III. §28).
COMPARISON OF THE TWO REPRESENTATIONS [132
Ex. v. In the expansion of the function ^ (x, a ; «, c) whose expression is given in
Example ii., the terms involving negative powers are
,- A _.
2s di ' Tc 2s clt't3'
and the terms involving positive powers are
where the quantities A, B, ..., E are those occurring in the expansion of y in the neigh
bourhood of the place c ; this expansion is of the form y = At + Bt3 + Ct» + Df + Et* + . . . .
Ex. vi. If in Ex. v. the integrals of the coefficients of t, t3 and t5 be denoted by
FI, F3Z, FZ, find the equation of the form
Fi=\Fi+p.F3* + integrals of the first kind + rational function of (s, 2)
which is known to exist (Chap. III. §§ 28, 26 ; Chap. V. § 57, Ex. ii.), X and p. being
constants.
Prove, in fact, if the surface be associated with the equation
7/2 = (x - cf +pl (X - C)5 + pz (X - C)4 +p3 (X - C)3 +Pt (X - Cf +pb (X - c)
that
^ + 2^?2 +Pi (z - c)J = - ^~Y + constant.
133. We pass now to a comparison of the two forms we have obtained
for each of the rational functions -^ (as, a; z, GI, ..., cp), ty(x, a; z, c), one
of which was expressed by the Riemann integrals, the other in explicit
algebraical form.
The cases of the two functions are so far similar that it will be sufficient
to give the work only for one case ijr (x, a ; z, clf ..., cp), and the results for
the other case.
From the two equations (§§ 122, 130)
•\Jr (x, a ; z, GI, . . . , Cp) = F*' a — 2, o)i (z) Tc'. ,
i=\
^(X,a: z,Cl,...,cp) = [(z,x)-(z,a)]-£-i «<(*)[(*, as)- (d, a)] £ ,
Ujli {,=1
we infer, denoting the function
byff*'a, that
1=1
133] OF THE FUNCTION A/r. 183
The function H*'a is not infinite at the place z, but is algebraically
infinite at infinity; it has the same periods as F*' a. The equation (ii) shews
that Hx'a I -y- is a rational function of z, while the equation
/ at
x a dz P
at j-=i ci
*' a
gives the form of F*' a / -^- as a rational function of z.
I at
Integrating the equation (iii) in regard to z, we obtain
p
nx, a, -r)Z, c . •*? TT-2, c ux> a /• \
z,c =Px,a+ ^ Vi Hc, (IV),
i=l
where c is an arbitrary place, and P^ °a is the integral of the third kind, as a
function of z, which was determined in Chap. IV. (§§ 45, 46).
Since the integral of the second kind E^'a, obtained in Chap. IV.
(§§ 45, 46), is equal to DzP*^", we deduce from the last equation, inter
changing x and z, and also a and c, and then differentiating in regard to z,
T?x'a _i_ f1 jrx>a n rjs'c r>Tis>c m-fx>a ri21'0 / \
&z +2, Kf X>z/fCi = JJgU^a, =DzIlZiC, =Ta (v),
and thence, using equation (iii) to express rx'a,
E*z> a = [(z, x} - (z, a}} | + £ [„, (z) H* a-Vfa DZH^ c] (vi),
which* gives the form of £%' / -£ as a rational function of ^.
/ ott
The difference of two elementary integrals of the second kind must needs be a function
which is everywhere finite, and therefore an aggregate of integrals of the first kind. The
equation (v) expresses the difference of E*' a and 1^' a in this way. But it should be
noticed that the coefficients of the integrals of the first kind in this equation, which
depend upon z, become infinite for infinite values of z. They are the quantities
D ffz'c.
Z Cj
From the equation (iv) we have
r,x,a_x,a P ^x.a.jZ.c
^.c-^c-.f^i Hc. >
wherein the coefficients of V?* on the right may be characterised as integrals of the
second kind. From this equation also, if the periods of V?* at thejth period loops of the
* An equation of this form is given by Clebsch and Gordan. Abel. Functnen. (Leipzig, 18G6)
p. 120.
184 ALGEBRAIC FORM OF INTERCHANGE [133
first and second kind be denoted by C^j and C'i,j respectively, we obtain, as the corre
sponding periods of P^' *
from these equations the periods of E*' a are immediately obtainable. These equations
may be used to express the integrals H*] ° in terms of the periods of f^ " at the period
loops of the first kind.
134. But all these equations are in the nature of transition equations;
they connect functions which are algebraically derivable with functions whose
definition depends upon the form of the period loops. We proceed further
to eliminate these latter functions as far as is possible, replacing them by
certain constants, which, in the nature of the case, are not determinable
algebraically.
The function of x expressed by H*'a is not infinite at the place z.
Hence we may define p2 finite constants A^ r by the equation
A D fTCr> c
A-iir — UCr O. c ,
where c is an arbitrary place. And if, as in § 132, Ex. in., we use the
algebraically determinable quantities given by
,. _. F, , dot] ,, (r. [, ^dc{ 1
Mi>r = DCr \(ci,cr)
we have
Mi,
and
Then, from equation (v), putting therein cr for z,
, a r-,x,a r, ^ , .-.dcr -nXya r/ x / , -, CtCr ^, . -TT-X, a / ••\
r =rcr -[(Cr,x)-(cr,a}]-~=ECr - [(cr,#)-(cr,a)] -^ + 2^ Ait r V\ (vn)
and thence, since Exc'r a = I dx DCr (x, cr)
J a
D*Hla = Der \(x, cr) d4\ - Dx \(cr,
at j )_
1=1
If in this equation we replace as by z and i by r and then substitute
in equation (v), we obtain
135] OF ARGUMENT AND PARAMETER. 185
and thus, if we define an, algebraically determinable, integral by the equation
ST.X, a r,x, a JJ i r<r,
Gz =EZ + SF
t=l
— \ 2 (M r< i — Mir) &)r (z) \ (viii),
r=l )
we have
rxz'a = Gxz'a + 2 7j'a 2 (^r)i + p/r f - p/t- r) &v (z),
i=l r=l
or
•n*. a
\ /\ / • • • \>
.)^^). (vm)>
i=l r=l
from which, by integration in regard to z, we obtain an equation
^.x,a fz ~x, a 7. rrX, a , r^"^.p / A A N T7-x, a Trz, e ,. x
&,C = ^2 ^=nZ)C-^ ss (Ar<i + Ai>r)Vi v; (ix),
*« f=i...p
either of these expressions being, by equation (viii), also equal to
WE, a- £ vj-x^l^ZtC r, v , ,n dc
Pz, c + F, j(5; - [(a, z)-(Ci, c)]
P P x a, -
i=l r=l
The equation (ix) shews that the integral Qx' a is such that
2t C %> &
while every term of (ix)' is capable of algebraic determination.
135. From the equation (ix), when none of the places x, z, d, ..., cp are
branch places, we obtain
dxdz dz^ "^i ' '
^ -^
i=l r=l
and hence, from the characteristic property ^— ^- Q*'* = ^-^-Q^l. we infer
r~P) ^ ~i r^ ci ~i^
c) [^'c<)-^i' ^r^t^^'^^Jl
f=l r=l
wherein every quantity which occurs is defined algebraically. The form
when some of the places are branch places is obtainable by slight modi-
186 CANONICAL INTEGRAL [135
fications. This is then the general algebraic relation underlying the funda
mental property of the interchange of argument and parameter, which was
originally denoted, in this volume, by the equation fiff * = II*' cft.
The relation is of course independent of the places clt ..., cp. For an expression in
which these places do not enter, see § 138, Equation 17.
The equation (xi) can be obtained in an algebraic manner (§ 137, Ex. vi.). The method
followed here gives the relations connecting the Riemann normal integrals and the particular
integrals obtained in Chap. IV., with the canonical integrals G^'a, Q*'".
It should be noticed, in equation (xi), that in the last summation each term occurs
twice. By a slight change of notation the factor £ can be omitted.
The interchange of argument and parameter was considered by Abel ; some of his
formulae, with references, are given in the examples in § 147.
136. From the equation (viii)' we have
r*»=0«.» + i |(^ii + ^i8)vf«.
C8 % i = l
From this equation, and the equation (viii)', we infer that
8=1 S=l
= ty(x, a; z, c1} ..., cp) (xii),
which result may be regarded as giving an expression of the function
ty (x, a ; z, d, ... , Cp) in terms of the integrals G ; but, written in the form
Gx' a = 2 a>s (z) Yx' a + [(z, x) - (z, a)] — - 2 a>i (z) [(ci} x) - (a, a)] -rr ,
S = l t*C i = l (Mi
the equation (xii) has another importance; if we call Q*'" an elementary
canonical integral of the third kind, and fl^ ., =DzQ*'", an elementary
canonical integral of the second kind, we may express the result in words
thus — The elementary canonical integral of the second kind with its pole at
any place z is expressible in the form
Z, ODS (z) Gx> a + (rational function of x, z, C1} . . . , cp) -r ,
s=i c» I &t
wherein the elementary canonical integrals occurring, have their poles at p
arbitrary independent places c1} . . . , cp.
Further, by equation (xii) the function E (x, z), of § 124, can be written
in the form
nx' a - 2 V'' c Gx' "
E(x,z) = e^'c -i c' (xiii).
137] AS FUNCTION OF ITS POLE. 187
If we put
K*'a=Gxz'a-[(Z,x}-(Z,a)}d^ /;" (xiv),
the equation following equation (xii) gives
p
rrX, a i? / \ rrX, a / \
K ' = 2 <Oi(z)K' (xv),
i=i ci
and therefore, also
Q*'°=P*'0+i VX'aK''e (xvi),
Z, C 2, C ^_i i Ct
and
which is another form of equation (xi).
It is easy to see that
137. Ex. i. Prove that the most general elementary integral of the third kind, with
its infinities at the places z and c, and vanishing at the place «, which is unaltered when
x, z are interchanged and also a and c, is of the form
.c.,
z=l r=l
wherein a,, r are constants satisfying the equations 0$, r=ar, f.
.Er. ii. If the integral of Ex. i. be denoted by ^ ", and Z), $£•" be denoted by Gx; a ,
prove that
iii. If, in particular, (£' a be given by
t, 0
..
i=l r=l
prove that
This is the integral, in regard to z, of the coefficient of ^ in the expansion of
\lr(x, a; z, c1} ..., cp), as a function of x, in the neighbourhood of the place ct (§ 132,
Ex. iii.).
The integral ^'" is algebraically simpler than the integral <^'ca, of this example, in
that its calculation does not require the determination of the limits denoted by M^ ;.
Ex. iv. For the case p = 1, when the fundamental equation is of the form
188 CALCULATION OF CANONICAL INTEGRAL. [137
if the variables at the place ct be denoted by x = clyy = dly the place not being a branch
place, prove that
and calculate §f ' a , from the equation xi, in the form
#i C
* °^ * 1 " dx • dz
where, if y*=f(x) = a()xi+4alx3 + 6a2x2 + <ia3x + ai, the symbol f(xyz) denotes the sym
metrical expression
a;2 («022 + 2a12+a2) + 2# («1^2 + 2a2z-f a3) + («2z2 + 2a32 + a4).
Prove also that in this case Mlt 1= —f (c1)/4/(c1).
Calculate the integral Qx" a when the place c: is a branch place, and prove that in that
case Mltlt =limitt=0(^ - +-), wherein x=cl + t2, y = At + Bt3+ ..., vanishes.
\-£i Ci — 30 t /
Ex. v. For the case (p = 2) in which the fundamental equation is
2
where f(x) is a sextic polynomial, taking cly c2 to be the branch places (cly 0), (c2, 0), in
the neighbourhood of which, respectively, x = c1 + t1yy y = A1tl + B1t13 + ... , and o;=
y — Azt2 + B2t23+..., prove that
and infer that
. . 1 dz 1
Supposing ^ and z have general positions, deduce from equation (ix) that
where ^x2, ^22 have been replaced by /' (c^, / (ca) respectively.
Prove that this form leads to
Q*,*= f fsy+f^^dxdz {* fx^dz M +
z'c Jcja 2(j«7-z)a y « JcJa2y2sL
where, if /(a?) be a0^ + 6a1^ + 15a2^4 + 20a3^3 + 15a4A-2 + 6a5^-|-a6, /(^, 2) denotes the
expression
and Z, JS/, ^V are certain constants depending upon ct and c2.
.Er. vi. Let R (x] be any rational function. By expressing the fact that the value of
the integral $R(x)dx taken round the complete boundary of the Riemann surface, is equal
137] UTILISATION OF THE FUNCTION \/r. 189
to the sum of its value taken round all the places of the surface at which the integral is
infinite, we shall (cf. also p. 232) obtain the theorem
where the summation extends to all places at which the expansion of R (x) -7- , in terms of
Cbt
[dx~\
R(x}~r- _l means the coefficient of
t~l in the expansion. If all the poles of R (x) occur for finite values of x, this summation
will contain terms arising from the fact that -j- contains negative powers of t when x is
infinite, as well as terms arising at the finite poles of R (x). If however R (x} be of the
form U(x) ^r V(x\ wherein U(x), V(x) are rational functions of x, whose poles are at
finite places of the surface, there will be no terms arising from the infinite places of the
surface.
Now let £ denote the current variable, and x, z denote fixed finite places : prove, by
applying the theorem to the case* when
&(€) = +(& «; z,cit ...,cp)^
that
D* f (xt z)-Dz + (z, x} = 2 {«, (x) |> (x,
< = 1 C{ ct
where ^(x, z) is written for shortness for ^(x, a; z, en ..., cp), and ty(x, z)]* denotes the
'*
coefficient of tc. in the expansion of ^ (x, z}, regarded as a function of x, in the neighbour
hood of the place c$.
Shew, when all the involved places are ordinary places, that this equation is the same
as equation (xii) obtained in the text.
Prove also that
Hence, as the forms a>f (.r) are also obtainable by expansion of the function ty (z, x), eveiy
term on the right hand is immediately calculable when the form of the function ^ (x, z)
is known ; then by integrating the right hand in regard to x and z we obtain an integral
of the third kind for which the property of the interchange of argument and parameter
holds. (Cf. Ex. iii. p. 180.)
Ex. vii. By comparison of the two forms given for the function ^(x, a; z, c) (§§ 126,
131), we can obtain results analogous to those obtained in §§ 133—136 for the function
+ (x,a; z,cl} ...,cp).
Putting, as before, H* a - I^1 ° - [(z, x) - (z, a)] -^ , and, when z is a branch place, under
standing by Z)*'1 H* a the expression D*z (n*'" - Pj cfl), and, further, putting
Gunther, Crelle, cix. p. 206.
190 MODIFICATION [137
wherein m is an arbitrary place and tc the infinitesimal at the place c, so that
^^-^B,,^^^^^:^^^
prove, in order, the following equations, which are numbered as the corresponding equa
tions in §§ 133—136 ;
<fl= JiW^"1^" (ii),
(vi),
wherein, when c is a branch place, the first term of the right hand is to be interpreted as
rff i px< a _ pc> m\ .
c \ c, m x, a) '
also the equations
,m - ' i ~c HC
i— i
+ I I
t'=l r=l
and thence, that the algebraically determinable integral
<r^"+j, T ^r [A (M *)-/>, (fe,)*
p p x a
•t=lr=l '" *
is equal to
I?*-|£ 1 V^'\r(z}(Br,i + Bi,r}
i=lr=l
and, finally, that the integral
_a;, a x, a P P ,.x,a ,,s,m
which, clearly, is such that &?**&mj can be algebraically defined by the equation
x, a J-.X, a
( i *r r i ' **rii (lx)'-
i r
Further shew that the function ^ (x, a ; z, c) can be written in the form
>//• (.r, a ; z, c) = G*' ' - 2 «»g (z) D* ff^' " (xii).
137] OF THE RESULTS OBTAINED. 191
The algebraical formula expressing the property of interchange of argument and parameter
is to be obtained from the equation
r> n f\x< a r» // \ °^\ 2 / \ rfi'1 ( r» // \ d>Z\ - /. . dc
DXD, ^ m=D. z) + „, (x) Zy A c) - A (', *)
+ \ 22 [a>i (a?) «,. (z) - <ar (a?) o^ (z)] JT, , r (x).
Lastly, if Lk(z) denote the coefficient of tk/\k (k positive) in the expansion of the function
•^ (x, a ; z, c) as a function of x in the neighbourhood of the place c, so that (Ex. iv. § 132)
where Pt, k denotes a certain constant such that P*,^ is Ni)r, prove, by equating to zero
the sum of the coefficients of the first negative powers of the infinitesimals in the expan
sions of the function of £, >//• (£, a ; z, c) D% ^ (£, a ; x, c), at all places where negative
powers occur, that
p
Dx ^ (x, a ; z,c)-D^(z,a; a?, c) = 2 [«< (a?) Lk. (z) - ^ (z) Lk. (x)] (A),
wherein, on the right, only functions Lk(z) occur for which k is one of the p numbers
^n ^2> •••» kp) and that
thus an elementary integral of the third kind, permitting interchange of argument and
parameter, is obtained immediately from the function ty(x, a; z, c) by integrating the
right hand of equation (B) in regard to x and z.
Prove also, that if
we have the formulae
p
•]lfi-lK^a (XV)
^r. viii. To calculate the integral Q*'^ for the case (p = 2) where the fundamental
equation is
2/2=/H,
wherein/^) is a sextic polynomial divisible by x-c, which is expansible in the form
•f(x} = A*(x-c} + Q(x-c? + R(x-cf + ...,
we may use the equation (xi) of Ex. vii. When x, z are near the place c, putting
prove that
D' \* ^ ^~t) ~ DX ^ x^ Jt = A* (** ~ ^2) + cubes and hiSher powers of «, and «2,
192 GENERAL STATEMENT [137
and thence (see Ex. ii. § 132) that
v r i \ i \ i \ , \-< R(x — z)dxdz
Kn [«! (*) 0>2 (Z) - o>2 (X) <*! (Z)] = V^ > -^ jt .
Also, when z is not a branch place, if Cj be a place near to c, and the expansion of the
function ^- (z, c^-^fa, 2) -^ in powers of the infinitesimal at c, contain the terms
M+ . + JVP + ... , so that
prove that
substituting these results in the formula (xi) of Ex. vii., prove that
_
z)z 240
where /(#, 0) has the same signification as in Example v. The part within the brackets
{ } is of the form ys^'2ai,r<ai(x)a>r(z), where ai,r = ar,i.
Obtain the same result by the formula (B) of Ex. vii., using the form of ^ (#, a ; z, c)
found in Ex. ii. § 132.
138. The formulae in §§ 133 — 136 enable us to express the form of a
canonical integral of the third kind, in the most general case ; and to
calculate the integral for any fundamental algebraic equation, when the
integral functions are known. But they have the disadvantage of presenting
the result in a form in which there enter p arbitrary places c1} ... , cp. We
proceed now to shew how to formulate the theory in a more general way ;
though the results obtained are not so explicit as those previously given,
they are in some cases more suitable for purposes of calculation.
Let u*' a, ... , ux'a denote any p linearly independent integrals of the first
kind ; denote Dxuf ° by /^ (#). Let the matrix whose (i, j)th element is
Hj (Ci) be denoted by p, c1} ..., cp being the places used (§ 121) to define
the quantities wl (x), ..., wp(x). Let i/ij denote the minor of the (i,j)ih
element in the determinant of the matrix /A, divided Jby the determinant
of /*; so that the matrix inverse * to p is that whose (i, j)ih element is Vjt{.
Then we clearly have
o>i 0) = Vi, i /*i O) + ...... + vi>PnP(x) (i=l, 2, ..., p).
* Since «*'", ..., «*'" are linearly independent, and the places clt ...,cp are independent
(see §§ 23, 121), the matrix /j.-1 can always be formed.
138] FOR FUNDAMENTAL INTEGRALS OF THE SECOND KIND. 193
Let a denote any symmetrical matrix of p2 quantities, a,-j, in which
aiij = Uji i. Then we define p quantities by the p equations
and call them fundamental integrals of the second kind associated with the
integrals u*'a, ... , u^". For instance when m (x) = &>; (x\ Vjj = Q unless
i=j, in which case *>,-,;=!. Thus by taking ait ; = £ (Ait } + A jti), the
integrals K*'a, ..., Kx'a (p. 187. xiv.) are a fundamental system associated
with the set V*'a, ..., V*'a.
It will be convenient in what follows to employ the notation of matrices
to express the determinant relations of which we avail ourselves *. We shall
therefore write the definition given above in the form
L*'*=vH*'a-2au*-",
wherein Lx'a stands for the row of p quantities Lx'a, ... , Lx'a, H'r'a stands
for the row of p quantities Hx'a, ... , Hx'a, and v denotes the matrix obtained
by changing the rows of v into its columns, and is in fact equal to the
matrix denoted by /z"1, so that we may also write
Lx' a = ^Hx' a
where (§137)
TT-t, a jr X.(t.tf*/t t .. TT%> ffl
Hci = Kc- + 2 S (Ar> i + Ai} r) Vr .
t
r=l
Explicit forms of the integrals K*'a have been given (§§ 134, 136).
Then, from the equations defining the integrals Lx.' a, we have
2 m (z) L? a = 2 Hxc'. a 2 i>j, ,: & (z) - 2 2 2 ar> , M *' " ps (z\
i=s\ * — i *•»' — i -n — i « _ i
, x TJ, n ,
= Z &>j (3rJ ±T; - 2 2 2 ar, g ur p* (z\
j=l r=l ,i=l
UX'a O <£ 4> a;'a / \
= •"« —22 2 ar) , wr ^, (2) ;
r=l s=l
and this is an important result. For, putting for z in turn any p independent
places, the p functions Lx'a are determined by this equation. Thus the
functions L^ a, ... t Lx' a do not depend upon the places c1 , c2, . . . , cp.
See for instance Cayley, Collected Works, vol. ii. p. 475, and the Appendix II. to the present
volume, where other references are given.
B- 13
194 CANONICAL INTEGRAL OF THE THIRD KIND. [138
Also, from this equation we infer
~ |~ . dz~\ j^ [, v<fe| n ITZ>C n TTx>a
x (*' X^~dt\~ * I ^ «8j * " ~
= I [^ (x) D,i? " - ^- (5) 0,1? a] (17),
1=1
c being any arbitrary place. Now it is immediately seen that if -Ri(#), ... ,
Rp (x) be any rational functions of x such that
then Ri (x) can only be a form of DXL? a, obtained from DXL*' a by altering
the values of the constant elements of the symmetrical matrix a. Hence
the equation (17) furnishes a method of calculating the integrals Lf", when
ever it is possible to put the left-hand side into the form of the right-hand
side.
The equation (17) shews that the expression
r> // \ da>\ , v / \ n r*- c
Dz ((x,z) -T. + 2 fit (x) DzLi ,
\ at/ i=i
is unaltered by the interchange of x and z. This expression is also
equal to
D, ((*, z) tg) + DZHX'C- 2 I iar^iir (*) ^ (z}
\ at/ r=\ s=i
and, -therefore, to
Hence, the formula (§ 134, ix.)
z> c
,.x, a r>x, a , x,arz,c -f-fX, a o . nl
Rg,c,=Pz,c + ^ ut Li =HZ|C -2 Z i ar>gw
i=l r=l s=l
_ a; « £, Ji / \ TT z. a TT-Z> c r» vr> ^ je, a z, c
= Qz c + k 2 2 (4r, , + -4,, r) 7r' F, - 2 2 S ar> sUr Us
r=l s=l r=l s=l
gives us a form of canonical integral of the third kind not depending upon
the places cl} ... , cpt and immediately calculable when the forms of the
functions Li are found.
The formula
If " = [(*, as) - (z, a)] + 2 fit (z) L*'a + 2 a,, . if * p. (z)
Clt ^=i r=\ s=l
serves to express any integral of the second kind in terms of the integrals
LI ..... Lv
139] EXAMPLE OF HYPERELLIPTIC CASE. 195
Ex. i. For the surface y2 =/(#)» where /(#) is a rational polynomial of order 2p + 2,
the function
^L_ d ( 9 N . * f f(& 2/tf)
'
wherein s2=f(z), »;2 =/(£), is a rational function of £ (without 17). Prove by applying the
theorem, 2 fj2(f) ^1 =0, (Ex. vi, § 137) that
where i; k' represent in turn every pair of unequal numbers from 0, 1, 2, ..., 2p, whose
sum is not greater than 2p, V being greater than k, and the coefficients X are given by the
fact that
Hence, a set of integrals of the second kind associated with the integrals of the first kind
/Q3C i OOQf\K t •
T7 ' / "17" » ' /
a, !/ J a 3 J a
y
is given by
x a fx dx fc = 2p+l-»'
Lfa= 2 Xt + l + f(*+l-i)«*f
J a *</ k = i
and a canonical integral of the third kind is given by
This is equal to
p+i
aJcZsfy (X-ZY
which is clearly symmetric in x and z.
O o
The value of 5- (z, x} — ^- (x, z) used in this example is given by Abel, (Euvres Completes
(Christiania, 1881), Vol. i. p. 49.
Ex. ii. Shew in Ex. i., for »=1, that the integral associated with I — is
J* y
fx X x -4- 2X x~
— dx ; and express these in the notation of Weierstrass's elliptic functions
J a *y
when the fundamental equation is y2=4.r3— gtfc-gy
139. Suppose now that the integrals u*'n, ...,u*'" are connected with
the normal integrals Vi'a, ... , v*'a by means of the equations
which, since H4- (x) = ZirHh*' n, are equivalent to
x,a
Ur =
Then the periods of the integral «*' a, at the first p period loops, form the
rth row of a matrix, 2\, and the periods of the integral ?^' " at the second
13—2
196 PERIODS OF FUNDAMENTAL INTEGRALS. [139
p period loops form the rib. row of a matrix 2Xr ; we shall write &> = X and
W' = XT, so that (t}jj=\ij. The two suffixes of the quantities ta^j will
prevent confusion between them and the differential coefficients o>i (x).
Let the periods of L^ at thejth period loops of the first and second kind
be denoted by — 2^ j and — Zrj'^ j respectively. The matrix whose t'th row
consists of the quantities 77^ i , ... , 77^ p will be denoted by 77 ; similarly the
matrix of the quantities 77'.^ will be denoted by 77'. The matrix of the
periods of the integrals H^a, ... , H^a at the first period loops is zero; the
(i, j)th element of the matrix at the second period loops is the jth period of
HC.' a, namely ft,- (Cj). We shall denote this matrix by A.
By the definitions of the integrals Lj'a we therefore have
and all these equations are contained in the equations
77 = 2a<u,
77' = 2am' - %v& = 2aw' - ^/t-1 A.
Now from the equations connecting f*,r (a) and fi, (as), we obtain
TTt'/i,. (Cj) = X,., ! f^! (C») + ...... + \r, p Op (Ci),
wherein /j,r (d) is the (i, r)ih element of the matrix /A, and the right hand is
the (i, r)th element of the matrix AX ; hence we may put
Trt'/i. = AX.
If then we denote the matrix ^/*-1A by h, we have
2AXA = 27ri/j.h = Tn'A = Avri,
and infer that 2XA = 7rt, and thence that 2/iX = iri. Thus 2hu>=Tri, 2ha)'=7rir.
A i , i • , •• *, a x, a x, a x, a ,11 i
Also tne integrals u-i , ... , up , ... , vi , ... , vp are connected by the
equation AM*- a = 2h\vx> a = Triif- a.
140. The four equations
2hw = iri, 2ha>' = TTIT, 77 = 2a&>, 77' = 2aay' — h (A)
will prove to be of fundamental importance in the theory of the theta
functions. They express the periods 77, 77' independently of the places
d, ... , cp, used in defining L*'a.
If beside the symmetrical matrix T, and the arbitrary symmetrical matrix
a, we suppose the matrix h, which is in general unsymmetrical, to be
141] RELATIONS CONNECTING THE PERIODS. 197
arbitrarily given, the integrals Ui' , ... , up' being then determined by the
equation hu*< a = 7rvix> a, the first equation, 2Ao> = TTI, gives rise to p2 equations
whereby the p* quantities wit j are to be found, and similarly the other
equations give rise each to p2 equations determining respectively the quantities
o>'i,j, rii,j, rfij- But, thereby, the 4p2 quantities thus involved are deter
mined in terms of less than 4>p2 given quantities. For the symmetrical
matrices a, T involve each only ^p(p + I) quantities, and the number of
given quantities is thus only p (p + 1) +p2. There are therefore, presumably,
4p2 — [pz +p (p + 1)], = 2p* —p,
relations connecting the 4p2 quantities (0,-j, co'ij, 77^-, ?/,;( j\ we can in fact
express these relations in various forms.
One of these forms is
corj = 770), W'TJ' = rj'o)', rjta' — wv) = ^TTI = rn'ij — TJ'CO, (B)
of which, for instance, the first equation is equivalent to the %p(p — 1)
equations
(wr, i IJr, j — tjr, i &> r, j) = 0,
r=l
in which i = 1, 2, . .. , p, j = 1, 2, ... , p, and i is not equal to j. The second
equation is similarly equivalent to ^p (p — 1) equations, and the third to p-
equations. The total number of relations thus obtained is therefore the
right number p- +p (p — 1), In this form the equations are known as
Weierstrass's equations.
Another form in which the 2p- — p relations can be expressed is
wo)' =&)'&>, 7777' = r}'rj, 0/77 — 0)77' = ^7ri = ijm' — t]'(a (C)
These equations are distinguished from the equations (A) as Riemanu's
equations.
141. The equations (B) and (C) are entirely equivalent; either set can be deduced
from the equations (A) or from the other set. A natural way of obtaining the set (B) is
to use the equation (17). A natural way of obtaining the set (C) is to make use of the
Riemaun method of contour integration.
The equations (A) give, recalling that a = a, a>' = a>7-, f = r,
o)ij = 2wa&) ,=/3, say, a symmetrical matrix,
<o»j' = 2u>r/o/ — w/i =
Hence ^ft)' = ^wr==^r=^T
and because w' = rw,
<5 V = TVTJ' = r/3r —
and thus, as r^r = T^T, \ve have
198 PROOFS OF THE RELATIONS [141
which are the equations (B). And it should be noticed that these results are all derived
from the three a>' = «r, urj =/3, &>;/=/3r — ^TTJ, assuming only that ft and T are symmetrical.
From the equations (B), putting »»;=£, uTj'=y, so that $ and y are symmetrical
matrices, we obtain*
?; = («5)~1jS, ^'=y(<o')~1, and thence ^'(w)"1^ — y(u)~1u = $ni,
Henue, if <O~IU>' = K, HO that O>K = O>', O>' = KO>, «' («) ~ 1 — *, and >c~1 = (a>')~1t<), we have
K/3 — yK~1 = ^7rl, or K/3K — y = ^7ri»c,
and therefore, as the matrices K/SK and y are symmetrical, so also is the matrix <c ; and thus
a> ~ l w' = a>' (o>) ~ *, and therefore ow' = <w'<o,
which is one of the equations (C).
Further
Wlf = tyw' — ^TTt = IJWK — ^Trt = /3K — JfTTZ,
and therefore q'w = icji - £TTI = K/3 - %iri\
leading to w^'w = £*£ - ^Trtft
and the right hand is a symmetrical matrix, and therefore equal to w^'/jtu ; thus also
W'|"9'f>
which is the second of the equations (C).
r inally (o> rj — w^') w = a)'jja) — <a((o'r] — ^TTI) — eo'tbjj
= ^-TT la),
and thus
to'^ -<»^' = ^7ri, = , therefore, r)H' — rj'oi,
which is the third of equations (C).
We have deduced both the equations (B) and (C) from the equations (A). A similar
method can be used to deduce the equations (B) from the equations (C).
Other methods of obtaining the equations (B) and (C) are explained in the Examples
which follow (§ 142, Exx. ii— v).
142. Ex. i. Shew that the^> integrals given by the equation
.x, a . fjx,a. .. Tix,a
Ai =tl>iHc, +--- + tPJi-acp >
where titj is the minor of Q,-^) in the determinant of the matrix A (§ 139), divided by the
determinant of A, namely by the equation
A*.« = A -!#*'",
are a set of fundamental integrals of the second kind associated with the set of integrals
of the first kind 2nivi' a, ... , ^irivf a, and are such that
'
= 1
.
^UWB.**')* 1 (ai(x)D,Kl° -<*>(,) DtK*'*
/ i=l \ J*
i=\
* The determinant of the matrix w, = X, cannot vanish, because u'", ..,?/'" are linearly
i /*
independent. The determinant of the matrix T does not vanish, since otherwise we could deter
mine an integral of the first kind with no periods at the period loops of the second kind
(cf. Forsyth, Theory of Functions, § 231, p. 440).
142] CONNECTING THE PERIODS. 199
Prove that the function A*' a has only one period, namely at the ith period loop of the
second kind, and that this period is equal to 1. For the sets
_ . x, a _ . x, a .x,a .x,a
2mv: , ..., 2mvp' , A,' , ..., A;) ,
we have in fact <a=iri, <a' = Trir, f? = 0, rj'=—^.
Shew that these values satisfy the equations (B) and (C).
Ex. ii. From Ex. i. we deduce
a • £• / '-'• a . *• c z,cx,a. & , x, a rz, c x,c Tx, a.
fcrt 1 (»< Aj -V. A« )= I («! Lt -Ut Li ).
i=\ i=\
Hence, supposing x and z separately to pass, on the dissected Riemann surface, respec
tively from one side to the other* of the rth period loop of the first kind, and from one
side to the other of the sth period loop of the first kind, we obtain, for the increment of
the right-hand side
p
-4 2 (&>i,ri7t,«-»7i,r<»t>«)>
i=i
which is the (r, s)th element of the matrix — 4 (GM; — i;a>). For the functions on the left-
hand side the matrix wrj — rjm vanishes (Ex. i.). Hence the same is true for those on the
right hand.
Supposing x to pass from one side to the other of the rth period loops of the first kind,
and z from one side to the other of the sth period loop of the second kind, we similarly
prove that 5»;' — ij<o' has the same value for the functions on the two sides of the equation,
and therefore, as we see by considering the functions on the left hand, has the value — fari.
While, if both x and z pass from one side to the other of period loops of the second kind
we are able to infer u'r)' = ^'<a'.
We thus obtain Weierstrass's equations (B).
Ex. iii. If Uf ",..., U*' a be any integrals, the periods of Uf a at thejth period loops
of the first and second kind be respectively &„-, f'f)>, and the matrices of these elements
be respectively denoted by £, f ' ; and W*'a , ..., W% a be other integrals for which the
corresponding matrices are £ and f , prove that the integral lU^d W*' a, taken positively
round all the period-loop-pairs has the value
which is the (i,j)th element of the matrix &'- f'f
Ex. iv. If #( (#) denote the rational function of x given by
p (fa
Ri (x)= 2 vr , » [(cr , a?) - (cr, a)] -^ ,
r=l
the function Zj**+jg|(*) is infinite only at c1? ...,cp, and has the same periods
Zr*' ° , Denote this function by Ya> a .
To that side for which the periods count positively (see the diagram, § 18).
200 SERIES REPRESENTING AN UNIFORM FUNCTION [142
Prove that if the expansion of the integral Y*' " in the neighbourhood of the place ct
be written in the form
then
</i,i = "i,i(^
where Ai, s, Mit t are as denned in § 134, and are such that Aitt + Mit t=Atti+Mtt t.
Hence shew that the sum of the values of the integral J r*'" dYf* taken round all
the places cx , . . . , cv is zero.
Ex. v. Infer from Exs. iii. and iv., by taking
/ \ 71^,0, ,.x. a iJT x, a ,-\ ,
(a) U. =u.' =W.' , that cow = « w,
fo\ frx,a-yx,a firx,a x, a ,-, , _. . ,
(P) U. = Y. , W. =u.' , that riS'-T)'5 = $iri,
/ \ Trx,a -irx, a Tirana fUr.4- v~' /-
(y) ™ =Y. W.' that 777 =J?'7.
These are Riemann's equations.
Ex. vi. If instead of the places cl} ..., cp and the matrix p., we use a matrix depending
only on one place c, the t'th row being formed with the elements D^~ ^ (c), ... , ZT' ^,(c),
we can similarly obtain a set Lf ",..., ^' a associated with the set «*' " , ... , u*' a .
Shew that the periods of Lx'a , ... , Zf' a thus determined are independent of the posi
tion of the place c.
Ex. vii. If the differential coefficients ^ O), ... , /*,,O), be those derived from a set of
p independent places bly b.^, ... , bp, just as ^ (x), ... , <ov(x) are derived from c^, ... , cp, so
that /*i(6i) = l, /ui(6,.) = 0, prove that i/,., i = co,. (6f) and that
143. We conclude this chapter with some applications* of the functions
•^r(x, a; z, c), E(x, z) to the expression of functions which are single-valued
on the (undissected) Riemann surface. Such functions include, but are
more general than, rational functions, in that they may possess essential
singularities.
Consider first a single-valued function which is infinite only at one place ;
denote the place by m, and the function by F (x).
dz
Since -\|r (x, a ; z, c) -^ is a rational function of z, the integral
/ CLt
\F(z}\^r (x, a • z, c) I -j- dz, or \F(z)-ty (x, a ; z, c) dtz,
taken round the edges of the period-pair-loops, has zero for its value. But
this integral is also equal to the sum of its values taken round the place m,
* Appell, Acta Math. i. pp. 109, 132 (1882), Giinther, Crelle cix. p. 199 ( 1892).
143] NEAR AN ESSENTIAL SINGULARITY. 201
where F(z} is infinite, and the places x and a at which -ty (x, a; z, c) is
infinite.
fJz
Now, when z is in the neighbourhood of the place m, since ilr (#, a ; z, c) I -=-
/ at
is a rational function of z, we can put
00 jf
^(x,a; z,c)- 2 J? Dr ty(x,a\ m, c),
r-O lr
where tm is the infinitesimal at the place m.
Thus the integral I F(z)ty(x, a ; z, c)dtz, taken round the place m, gives
oo Ar
Zjri 2 -i-f Drm -f O, a ; m, c\
?' = 0 |_
1 f
where J.r is the value of the integral — . \trmF(z}dtz taken round the
place m.
When 2 is in the neighbourhood of the place x, i/r (x, a; z,c) is infinite
like tx , tx being the infinitesimal at the place as, and therefore, taken round
the place x, the integral
J F(x)^f (x, a; zt c)dtz
gives
Similarly round the place a, the integral gives — 27riF(a).
Hence the function F(x) can be expressed in the form
F(x) = ^(a) - J^ T^' Drm ^ (x, a ; m, c),
the places a and c being arbitrary (but not in the neighbourhood of the
place m).
For example, when p = 0, ^ (x a; z, c)= - ( — --- L \
\& — z a-zj
an(J
wherein
1 /"
^r = 2nri / (2 — m)r f(z} dz, the integral being taken round the place m.
A similar result can be obtained for the case of a single valued function with only a
finite number of essential singularities. When one of these singularities is only a pole,
say of order M, the integral /£ F(z) dz, taken round this pole, will vanish when r5M, and
the corresponding series of functions Dr ^ (.>-, a • m, c) will terminate.
202 MITTAG LEFFLER'S THEOREM [144
144. We can also obtain a generalization of Mittag Leffler's Theorem.
If c1; c2, ... be a series of distinct places, of infinite number, which converge*
to one place c, and f± (x), f2 (a), ... be a corresponding series of rational
functions, of vnuca/V(0) is infinite only at the place Cj, then we can find a
single valued function F (x), with one essential singularity (at the place c),
which is otherwise infinite only at the places cit C2> ..., and in such a way
that the difference F(x) —J\ (x) is finite in the neighbourhood of the place Cf.
Since fi(x) is a rational function, infinite only at the place C{, and
•fy(x,a; x, c) does not become infinite when z comes to c, we can put
ft («) =/* (a) ~ 2 -f Dl + (x, a • a, c), (A)
r=0 ;_L
wherein a is an arbitrary place not in the neighbourhood of any of the
places Cj, c2, ..., c, and \ is a finite positive integer, and Ar a constant.
Also, when z is sufficiently near to c, and x is not near to c, we can put
«, tk
TJT (as, a ; z, c) = 2 « [D^ ^r (ar, a ; s, c)]z=c,
fc=0 \K
wherein tc is the infinitesimal at the place c. Thus also, when z is near to c,
17 ^ (x, a • z, c) m |o tkc Rk (as), (B),
wherein Rk (x} is a rational function, which is only infinite at the place c.
There are p values of k which do not enter on the right hand ; for it can
easily be seen that if &!,..., kp denote the orders of non-existent rational
functions infinite only at the place c, each of the functions
[D*'->O, a; z, cXU, , [JOf-1 ^ (*, a ; z, c],=c
vanishes identically. Let the neighbourhood of the place c, within which z
must lie in order that the expansions (B) may be valid, be denoted by M.
Of the places Cj , c2 , . . . , an infinite number will be within the region M ;
let these be the places cs+l, cs+.2, ...: then s will be finite and, when i > s,
we have
oo £
&'. ty (*', a; Ci,c)= S << ^ * (as),
k=0
wherein ti is the value of tc, in the equation (B), when z is at c;. Hence also,
from the equation (A), wherein there are only a finite number of terms on
the right hand, we can put
fi(x}-fi(a)= I £$,*(*), (C),
k = Q
wherein S^k is a rational function, i > s, and a; is not near to the place c.
* so that c is what we may call ike focus of the series c1} c3, ... (Haufungsstelle).
144] FOR ANY UNIFORM FUNCTION. 203
It is the equation (C) which is the purpose of the utilisation of the
function ty (x, a ; z, c) in the investigation. The functions S^k (#) will be
infinite only at the place c. The series (C) are valid so long as x is outside
a certain neighbourhood of c. We may call this the region M'.
Let now eg+i, eg+2, ... be any infinite series of real positive quantities, such
that the series
C8+l H~ e«+2 + €S+3 + • • •
is convergent ; let ^ be the smallest positive integer such that, for i > s, the
terms
00 k
2 ti $,:, jfc (x),
fc=c*i+l
taken from the end of the convergent series (C), are, in modulus, less than et-,
for all the positions of x outside M ' ; then, defining a function gt (x), when
i > s, by the equation
^ k
9i O) =/i OB) ~fi (a) ~ 2 tt Si, k (a;),
fc=0
we have, tor i > s,
Thus the series
2 [/((*)-/<(«>] + 2
t=l i=s+l
is absolutely and uniformly convergent for all positions of x not in the
neighbourhood of the places c, clt c2, ..., and represents a continuous single
valued function of x. When a; is near to cit the function represented by the
series is infinite like /f (#).
The function is not unique ; if i/r (x) denote any single-valued function
which is infinite only at the place c, the addition of ty (x) to the function
obtained will result in a function also having the general character required
in the enunciation of the theorem. As here determined the function
vanishes at the arbitrary place a ; but that is an immaterial condition.
For instance when p = 0, and the place m is at infinity, the places mlt m2, «ia, ...,
being 0, 1, w, l+a, ... , p + qa>, ... , wherein o> is a complex quantity and pt q are any
rational integers, let the functions f^x), /,(#), ... be x~\ (x-l}~\ (x-m)-\ ... ,
He, B c=
when z is great enough and | x\ < \ z |, | a \ < \ z |.
Also
1 _ ix-a
a-nti \mS
when im is great enough, and | x \ < \ m < | , | a \ < \ m^ \ .
204 FORMATION OF AN UNIFORM FUNCTION [144
Now the series
2 ^ =22
mf
is convergent. Hence when x and a are not too great
'
where et is a term of a convergent series of positive quantities. This equation holds for
all values of i except i = l, in which case mi = 0.
Hence we may write
x-mi a-mi mf
and obtain the function
i r — i i x~a i
=-ao \_x-p-qo a-p-qe> (p + qu^J*
* «• p= — oo q
which has the property required. This function is in fact equal, in the notation of
"VVeierstrass's elliptic functions, to f(.t- 1, a>) — f(a 1, «).
145. We can always specify a rational function of x which, beside being
infinite at the place c, is infinite at a place d like an expression of the form
T-° + -2-1+ +
tci %
ci
namely, such a function is
and this may be used in the investigation instead of the function fi (x} —fi (a).
Hence, in the enunciation of the theorem of § 144, it is not necessary
that the expressions of the rational functions fi (x) be known, or even that
there should exist rational functions infinite only at the places Cj in the
assigned way. All that is necessary is that the character of the infinity
of the function F, at the pole C;, should be assigned.
Conversely, any single-valued function F whose singularities consist of
one essential singularity and an infinite number of distinct poles which
converge to the place of the essential singularity, can be represented by
a series of rational functions of x, which beside the essential singularity have
each only one pole.
146. Let the places c1} c2, ..., c be as in § 144. We can construct a
single-valued function, having the places c1; c2, ..., as zeros, of assigned
positive integral orders Xj, X2, ..., which is infinite only at the place c, where
it has an essential singularity.
147]
WITH GIVEN ZEROS.
205
For the function
E (x, z} -
is zero at the place z and infinite only at the place c. When z is near to c
we can put
- tr
Dz log E (x,z) = 2 f [DJ ^ (*, a ; *, c)]z=c,
r=0
and therefore, when c4- is near to c, and # is not near to the place c, we
can put
\t\ogE (a, a) = 2 tJJBi,*(*),
fr=o
wherein R^jg^ae) is a rational function of x which is infinite only at the
place c, and ti has the same significance as in § 144.
Let the least value of i for which this equation is valid be denoted by
s+ I, and, taking e,+1, es+2, ... any positive quantities such that the series
is convergent, let /*t- be the least number such that, for i > s,
00 Jf-
2* vi JTVV Jf \^ ) ^ ^1*
Then the series
* 00 / V-f
2 Xi log E (x, d) +2 ( \i log E (x, mi) — 2 ti R^ k (x.
i=l i=s-fl \ fr = 0
consists of single-valued finite functions provided x is not near to any of
GI, c2, ..., c, and, by the condition as to the numbers //*, is absolutely and
uniformly convergent.
Hence the product
« 00
n \E (x col** n
represents a single-valued function, which is infinite only at c where it has
an essential singularity, which is moreover zero only at the places c1} C2, ...
respectively to the orders \lt \2,
With the results obtained in §§ 144 — 146, the reader will compare the
well-known results for single-valued functions of one variable (Weierstrass,
Abhandlungen aus der Functionenlehre, Berlin, 1886, pp. 1 — 66, or Mathem.
Werke, Bd. ii. pp. 77, 189).
147. The following results possess the interest that they are given by Abel ; they
are related to the problems of this chapter. (Abel,' (Euvres Completes, Christiania, 1881,
vol. i. p. 46 and vol. ii. p. 46.)
206 INTERCHANGE OF ARGUMENT AND PARAMETER. [147
Ex. i. If $ (x) be a rational polynomial in x, =n
and / (.r) be a rational function of x, =
then
-/(») <6 (a?) ,
The theorem can be obtained most directly by noticing that if 0 (#, 2) =
<p (z) (x — z)
then
is a rational function of X Denoting it by R(X} and applying the theorem
we obtain Abel's result.
Ex. ii. With the same notation, but supposing f(x} to be an integral polynomial,
prove that
wherein Akt1c't is a certain constant, and \^ (x) is the product of all the simple factors of
This result may be obtained from the rational function
as in the last example.
Ex. iii. Obtain the theorem of Ex. ii. when f(x)=Q, and 0 (x) = ty (.>•)]'"• In the
result put »i=— £, and obtain the result of the example in § 138.
These results are extended by Abel to the case of linear differential equa
tions. Further development is given by Jacobi, Crelle xxxii. p. 194, and by
Fuchs, Crelle Ixxvi. p. 177.
.148]
CHAPTER VIII.
ABEL'S THEOREM; ABEL'S DIFFERENTIAL EQUATIONS.
148. THE present chapter is mainly concerned with that theorem with
which the subject of the present volume may be said to have begun. It will
be seen that with the ideas which have been analysed in the earlier part of
the book, the statement and proof of that theorem is a matter of great
simplicity.
The problem of the integration of a rational algebraical function (of a
single variable) leads to the introduction of a transcendental function, the
logarithm ; and the integral of any such rational function can be expressed
as a sum of rational functions and logarithms of rational functions. More
generally, an integral of the form
\
dxR(x,y, y,, ..., yk),
wherein x, y, yl} yz, ... are capable of rational expression in terms of a single
parameter, and R denotes any rational algebraic function, can be expressed
as a sum of rational functions of this parameter, and logarithms of rational
functions of the same. This includes the case of an integral of the form
\dx R (x, Vatf2 + bx + c).
But an integral of the form
\dx R (x, Vow?4 + bxs + cx2 + dx + e)
cannot, in general, be expressed by means of rational or logarithmic functions ;
such integrals lead in fact to the introduction of other transcendental func
tions than the logarithm, namely to elliptic functions ; and it appears that
the nearest approach to the simplicity of the case, in which the subject
of integration is a rational function, is to be sought in the relations which
exist for the sums of like elliptic integrals. For instance, we have the
equation
I"*1 j dx jx* dx_ [x* (/.'.' _ .,
h */(l-tf\(1—tMf\ o A/H -.7? VI -kW Jo \/Cl -ar'Ul - k*a*)
208 INTRODUCTORY. [148
provided
On further consideration, however, it is clear that this is not a complete
statement ; and it is proper, beside the quantity x, to introduce a quantity y,
such that
and to regard y, for any value of x, as equally capable either of the positive
or negative sign ; in fact by varying x continuously from any value, through
one of the values x=±\, x=±j, and back to its original value, we can
K
suppose that y varies continuously from one sign to the other. Then the
theorem in question can be written thus ;
/•<*- 0>> dtKi [<*" ^ dx2 [<**• y^dxs_
l(o, i) 2A J(o, i) 2/2 J (o, i) 2/3
where the limits specify the value of y as well as the value of x. The
theorem holds when, in the first two integrals the variables (x, y) are taken
through any continuous succession of simultaneous values, from the lower to
the upper limits, the variables in the last integral being, at every stage of
the integration, defined by the equations
ys (1 - l&xfxff = S/iJ/2 (1 + AfteiW) - XjX% (I - &asfaif 1 - 2.
The quantity y is called an algebraical function of x\ and the notion thus
introduced is a fundamental one in the theorems to be considered ; its
complete establishment has been associated, in this volume, with a Riemann
surface.
In the case where y2 = (1 — #2) (1 — &2#2) we have the general theorem
that, if R (x, y} be any rational function of x, y, the sum of any number, m,
of similar integrals
f(*i, I/,) r^m' 2/m>
M (a, y) dx + ...... + 1
J«-..*0 **
R(x,y)dx
can be expressed by rational functions of (x1} y^, ..., (xm, ym), and logarithms
of such rational functions, with the addition of an integral
X*m+i. y«+i>
R (x, y} dx.
•/(««+!> ZWl)
Herein the lower limits (c^, b^, ..., (am, bm) represent arbitrary pairs of
corresponding values of x and y, and the succession of values for the pairs
(a?i, 2/j), ..., (xm, ym) is quite arbitrary; but in the last integral a'm+1, yM+i are
each rational functions of (scl} y^, ..., (xm, ym), which must be properly deter-
loO]
INTRODUCTORY STATEMENT.
209
mined, and it is understood that the relations are preserved at all stages of
the integration, so that for example am+l, bm+1 are respectively taken to be
the same rational functions of (a1} 6j), ..., (am, bm). The question of what
alteration is necessary in the enunciation when this convention is not
observed, is the question of the change in the value of an integral
+l> ?/m+l>
R (x, y) dx
+l> &m+l>
when the path of integration is altered. This question is fully treated in the
consideration of the Riemann surface, with the help of what have been called
period loops.
149. Abel's theorem may be regarded as a generalization of the theorem
just stated, and may be enunciated as follows : Let y be the algebraical
function of x defined by an equation of the form
f(y, x) = yn+A}yn-> + ...... + An = 0,
wherein Aly . .., An are rational polynomials in cc, and the left-hand side of
the equation is supposed incapable of resolution into the product of factors of
the same rational form ; let R (x, y) be any rational function of x and y ;
then the sum of any number, m, of similar integrals
r(*i,3/i> n
I R(x,y)dx + ...... + I
R (x, y) dx,
with arbitrary lower limits, is expressible by rational functions of (xlt ;?/,), ...,
(xm, 2/m)> and logarithms of such rational functions, with the addition of the
sum of a certain number, k, of integrals,
— I R (x, y) dx — — I
R (x, y) dx,
wherein zl} . .., zk are values of x, determinable from xlt yl} ..., xm, ym as the
roots of an algebraical equation whose coefficients are rational functions of
x\> y\> •••, ®m, ym, and sl} ..., Sfc are the corresponding values of y, of which
any one, say S{, is determinable as a rational function of zt, and aelt ylt ...,
xm, ym- The relations thus determining (zl} s^, ..., (zk, sk) from (x1} y^, ...,
(xm, ym} may be supposed to hold at all stages of the integration ; in
particular they determine the lower limits of the last k integrals from the
arbitrary lower limits of the first m integrals. The number k does not
depend upon m, nor upon the form of the rational function R (x, y} ; and in
general it does not depend upon the values of (#,, y^, ..., (>„,, ?/,„), but only
upon the fundamental equation which determines y in terms of x.
150. In this enunciation there is no indication of the way in which the
equations determining zly sl} ..., zk, sk from xlt ylt ..., xm, ym are to be found.
Let 6 (y, x) be an integral polynomial in x and y, wherein some or all of the
coefficients are regarded as variable. By continuous variation of these
B. 14
210 STATEMENT OF ABEL'S THEOREM. [150
coefficients the set of corresponding values of x and y which satisfy both
the equations f(y, x) = 0, 6 (y, x) = 0, will also vary continuously. Then, if
m be the number of variable coefficients of 6 (y, x}, and m + k the total
number of variable pairs (x, y} which satisfy both the equations f(y, x} = 0,
Q(y, x) = 0, the necessary relations between (x^, y^), ..., (xm, ym), (zl} s^), ...,
(zjf, Sk) are expressed by the fact that these pairs are the common solutions of
the equations /(y, x) = 0, 6 (y, x) = 0. The polynomial 6 (y, x} may have any
form in which there enter m variable coefficients ; by substitution, in 6 (y, x},
of the m pairs of values (xlt y^, ..., (xm, ym), we can determine these variable
coefficients as rational functions of xlt y1} ..., xm, ym\ by elimination of y
between the equations 6 (y, x} = 0, f(y, x) = 0, we obtain an algebraic equa
tion for x, breaking into two factors, P0 (x) P (x) = 0, one factor, P0 (x), not
depending on xlt ylf ..., xm, ym, and vanishing for the values of x at the
fixed solutions of f(y,x) = Q, 0(y,x) = 0, which do not depend on x,,ylt
..-, xm, ym, the other factor, P (x), having the form
(x-xj ...(x-xm)(xk + R^-1 + ... + Rk),
where Rly ..., RK are rational functions of x1} ylt ..., xm, ym. Finally, from
the equations /(>;, zi) = Q> 0(si, z^ = Q we can determine s-i rationally in
terms of zit xlt yly ...,#,», ym. As a matter of fact the rational functions of
#1, 2/i> ••• > xm, ym, which appear on the right-hand side of the equation which
expresses Abel's theorem, are rational functions of the variable coefficients in
151. When 0(y,x) is quite general save for the condition of having
certain fixed zeros satisfying f(y, x) = 0, the forms of (zly sj, ..., (zky sk) as
functions of (xlt y^, ...,(xm> ym) are independent of the form of 6 (y, x}. This
appears from the following enunciation of the theorem, which introduces
ideas that have been elaborated since Abel's time, and which we regard as the
final form — Let (a1; 6j), ...,(aQ, bQ) be any places of the Riemann surface
whatever, such that sets coresidual therewith have a multiplicity q, and a
sequence Q — q=p — r—l, where r + I is the number of 0 polynomials
vanishing in the places (a^ 6j), ..., (aQ, bQ); let (a?,, y^, ..., (xq, yq} be q
arbitrary places determining a set coresidual with (alt 6j), ..., (aQ, 6C), and
(z1} Si), ..., (ZP-T-I, SP-T-I) be the sequent places of this set* ; then, R (x, y)
being any rational function of (x, y), the sum
n*uj/i) /•(*«• v,) _
R(x,y)dx+ ...... + R(x,y)dx
J (a,, V 1 (a?, 6«)
is expressible by rational functions of (xlt y^, ..., (xq, yq}, and logarithms of
such rational functions, with the addition of a sum
/•<?!, «i> riv-i.^-T-i)
— R(x,y}dx— ...... — R (x, y) dx
• ' («»+i, b«+i) J (as, b^
* See Chap. VI. § 95.
152] REDUCTION TO TWO SIMPLE CASES. 211
herein it is understood that the paths of integration are such that at every
stage the variables form a set coresidual with (alt 6j), ..., (aQ, bQ).
The places (aly b^, ...,(cig, bQ) may therefore be regarded as the poles, and
(a?i, ^j), ..., (xq, yq), (X, Sj), ..., (^p_T_j, .Sp_T_!) as the zeros, of the same rational
function Z (x) ; if dl (y, x) denote the form of the polynomial 0 (y, x) when it
vanishes in (a1} bj, ..., (aQ, bq), and 0%(y, x} denote its form when its zeros
are (xlt y^, ..., (zlt s^, . .., the function Z (x) may be expressed in the form
#2 (y, x}jdi(y, x}. If the polynomials 0i(y, x}, 0%(y, x) are not adjoint, the
function will be of the kind, hitherto regarded as special, which takes the
same value at all the places of the Riemann surface which correspond to a
multiple point of the plane curve represented by the equation f (y, x) = 0 ;
this fact does not affect the application of Abel's theorem to the case.
152. To prove the theorem thus enunciated, with the greatest possible
definiteness, we shew first that it may be reduced to two simple cases.
In the neighbourhood of any place of the Riemann surface, at which t is
the infinitesimal, we can express R (x, y)-r. in a series of positive and
negative powers of t, in which the number of negative powers is finite. Let
the expression at some place, £, where negative powers actually enter, be
denoted by
then, if P£ £ denote any elementary integral of the third kind, with infinities
at f, 7, and E*' c denote the differential coefficient of P^' c in regard to the
infinitesimal at £, the places 7, c being arbitrary, the difference
wherein D^ denotes differentiation in regard to the infinitesimal at £, is finite
at the place £ The number of places, £, at which negative powers of t enter
• dec
in the expansion of R (x, y) -=- , is finite ; dealing with each in turn we obtain
an expression of the form
wherein 7, c are taken the same for every place £ ; this is finite at all places
of the Riemann surface, except possibly the place 7. If ty be the infinitesi
mal at this place the function is there infinite like (2AJ log ty. But in fact
S4, is zero (Chap. II. § 17, Ex. (S): Chap. VII. § 137, Ex. vi.). Hence the
14-2
212 PROOF OF THE THEOREM. [152
function under consideration is nowhere infinite, and is therefore necessarily*
a linear aggregate of integrals of the first kind, plus a constant. Hence
if ua> a, ..., ua'' " be a set of linearly independent integrals of the first kind, a
denoting the place (a, 6), and (7j , . . . , Cp be proper constants, we have
The consideration of the sum
r*i r*u
i R(x,y}dx + ...... + 1 R(x,y)dx,
J tt a
wherein a^, ..., aQ denote the places (al} 6j), ..., (a^, bQ), and xlf ..., XQ denote
the places (xly yj, ..., (xq, yg), (z1} s^, ..., (zp^_lt sp-^-^), is thus reduced to
the consideration of the two sums
,, ...
Ex. i. By the proposition here repeated from § 20, Chap. II., it follows that any
rational function can be written in the form
+ (^, I/-1'1 0B_1 (x, y)]/f (y)
where (cf. § 45, Chap. IV.)
n-l
(x, |) = [<^o (^ y) H- 2 0,. (a?, y) gr (£, r,)]/(x - £)f (y),
i
»; being the value of y at the place £.
.£"#. ii. Prove also that any rational function with simple poles at |1} £2, ... can be
written in the form
Xj, A2>"' bein§ constants, and « denoting an arbitrary place (cf. § 130, Chap. VII.).
153. We shall prove, now, in regard to these two sums, under the
conventions that the upper limits are coresidual with the lower limits, and
that the Q paths of integration are such that at every stage the variables are
at places also coresidual with the lower limits, a convention under which the
paths of integration may quite well cross the period loops on the Riemann
surface, that the first sum is zero for all values of i, and the second equal to
log Z(£)/Z(ry), Z(oc) being the-}- rational function which has alt ..., aQ as
poles and x1} ..., x^ as zeros. The sense in which the logarithm is to be
understood will appear from the proof of the theorem. If we suppose the
lower limits arbitrarily assigned, the general function Z (x), of which these
* Forsyth, Theory of Functions, § 234.
t If two rational functions have the same poles and the same zeros their ratio is necessarily
a constant.
154] PROOF OF THE THEOREM. 213
places a1( ... , aq are the poles, will contain q 4- 1 arbitrary linear coefficients,
entering homogeneously, and the assignation of q of the zeros, say x1, ...,xq,
will determine the others, as explained. — The equations giving the determi
nation will be such functions of a,, ..., aQ as are identically satisfied by these
places, Oj, ..., (tq. Hence the general form of Abel's theorem is
where Z' (f ) = D$Z (£) ; the term 2-4j log Z (7) = log Z (7) 2-4 1 can be omitted
because 2^ = 0 (Chap. II. p. 20 (8)). Herein Z '(£) is a rational function of
n n Q nn 'T* o*
lt/i j . . . . \AJQ CvlJ.VA lA/i j * • • j t^rt •
154. In carrying out the proof we make at first a simplification — Let
Z(x), or Z, be the rational function having a1} ..., aQ as simple poles and
#!, ..., #c as simple zeros, these places being supposed to be all different;
trace on the Riernann surface an arbitrary path joining at to x^ chosen so as
to avoid all places where dZ is zero to higher than the first order, and let /j,
be the value of Z at any place of this path ; then there will be Q — 1 other
places at which Z has the same value JJL ; the paths traced by these Q — 1
places as /* varies from oc to 0 are the paths we assign for the Q — 1 integrals
following the first. The simultaneous positions thus defined for the variables
in the Q integrals are, for q > 1, not so general* as those allowed by the con
vention that the simultaneous positions are coresidual with Oj, ..., UQ; but it
will be seen that the more general case is immediately deducible from the
particular one.
Consider now, for any value of JJL, the rational function
1 dl
Z-fji dx'
I, = IR(x, y)dx, being any Abelian integral whatever. In accordance with
a theorem previously used (Chap. II. p. 20 (8) ; Chap. VII. § 137, Ex. vi.) the
sum of the coefficients of t~l in the expansions of (Z - n)~ldlldt, in terms of
the infinitesimal t, at all places where negative powers of t occur, is equal to
zero. Of such places there are first the Q, places where Z is equal to //.. We
shall suppose that dl/dt is finite at all these places ; then the sum of the
coefficients of t~l at these places is
(^ ;fdA M
* Sets coresidual with two given coresidual sets have a multiplicity q; but sets equivalent
with two given coresidual sets have a variability expressible by one parameter only (cf. Chap. VI.
§§ 94-96).
214 PROOF OF THE THEOREM. [154
provided Z — //, be not zero to the second order at any of the places, that is,
provided dZ be not zero to higher than the first order. In accordance with
the convention made as to the paths of the variables in the integrals, we
suppose this condition to be satisfied.
Hence this sum is equal to the sum of the coefficients of t~l in the
expansions of the function - (Z - /Lt)"1 dl/dt at all places, only, where dl/dt is
infinite; this result we may write in the form
H
we may regard this equation as a convenient way of stating Abel's theorem
for many purposes; and may suppose the case, in which an infinity of dl/dt
coincides with a place at which Z = p, to be included in this equation, the
left hand being restricted to all places at which Z = p and dl/dt is not
infinite.
In this equation, in case I, = u*'a, be any integral of the first kind, the
right hand vanishes; then, integrating in regard to //, from oo to 0, we
obtain
In case / be an integral of the third kind, = Pf c say, and Z be not equal to
fi either at £ or 7, the right hand is equal to
1 1
to\ I
hence, integrating,
Z>»1, «i , >pXQ'ali - rj.. - lr>». /m
pt. ' -d*- + ' g'
while, if the places at which the rational function Z (x) has the values /JL, v be
respectively denoted by
•&! } ...... ) «£ Q)
and
&l> ...... ) a Q
we have
pz,', a/ p*tt,a'u _ ["• 7 / _L __ _1
^ 5^ L*P\ ZM-^ZM-
For any Abelian integral we similarly have
/- • a> + ...... + 1** ?•
which is a complete statement of Abel's theorem.
155] REMARKS. 215
155. In the equation (B), and in the equation which follows it, the
significance of the logarithm is determined by the path of /j, in the integral
expression which defines the logarithm ; we may also define the logarithm by
considering the two sides of the equation as functions of f.
There is no need to extend the equation (B) to the case where one of the
paths of integration on the left passes through either £ or 7, since in that
case a corresponding infinite term enters on both sides of the equation.
But it is clear that the condition that no two of the upper limits xl, ... , #g
should be coincident is immaterial, and may be removed. And if two (or
more) of the places at which Z takes any value, /j,, should coincide, the
equations (A) and (B) can be formed each as the sum of two equations in
Avhich the course of integration is respectively from Z — GO to Z= /JL and from
Z = fj, to Z = 0, and the final outcome can only be that the order in which the
upper limits xl, ..., XQ are associated with the lower limits a^ , . . . , aQ may
undergo a change. But in the general case we may equally put, for example,
in equations (A), (B),
/•*i fz2 r*2 f*i C^ . /•«, rx2 rxl
dl + dl,= dl+ dl+ dr+ dl,= dl+ dl,
J al J a2 J a, J jr2 •* x\ J at J a, •' a2
with proper conventions as to the paths ; hence the condition that dZ shall
not be zero to higher than the first order at any stage of the integration may
be discarded also, with a certain loss of definiteness. The most general form
of equation (A), when each of the Q paths of integration are arbitrary, is of
course
iiP + Ml'»titl + ...... + Mp'a>'i>p, (C)
where w^, ..., w'i>p are the periods of ufa and Mlf ..., Mp' are rational
integers, independent of i. We shall subsequently see that this equation is
sufficient to prove that the places xl} ..., XQ are coresidual with the set
a,, ..., aQ.
If, in equation (B), we substitute for Z(x) any one of its rational
expressions, say* 6z(x)ldl(x), we shall obtain
where, now, 02(x), O^x) are any two polynomials, integral in x and y, of
which, beside common zeros, 02(x) has xlt ..., xq for zeros, and 6^(x) has
OL •••, «y for zeros. If in this equation we suppose any of the coefficients in
Q* (x) to vary infinitesimally in any way, such that the common zeros of 02 (x}
9 (x) is, for shortness, put for what would more properly be denoted by 9 (y, x).
CASE IN WHICH THE LOGARITHM [155
and $!(&•) remain fixed, #2(#) changing thereby into 02 (x) + 802 (x), the places
AI, ..., xq changing thereby to xl + dxl} ..., XQ + dxQ> we shall obtain
= S log
^- ,
02 (7)
Avhich is slightly more general than any equation before given, in that the
places Xi + dxi, ..., xQ+dxQ, though coresidual with xlt ..., XQ, are not
necessarily such that" the function 0a (#)/0, (a;) has the same value at all of
them. This general equation is obtained by Abel in the course of his proof
of his theorem.
For any Abelian integral we have, similarly, the equation
which, also, may be regarded as a complete statement of Abel's theorem.
156. In equation (B) the logarithm of the right hand will disappear if
= z("t\ namely if the infinities of the integral be places at which the
function Z (x) has the same value.
One case of this may be noticed ; if ^ (?/, x) be an integral polynomial of
grade (•/* - 1) a + n - 3 (cf. Chap. VI. §§ 86, 91), which is adjoint at all places
except those two, say A, A', which correspond to an ordinary double point of
the curve represented by the equation f(y, x) = 0, the integral
/•/ / \ "**'>
/ (y)
will be an integral of the third kind having A, A' as its infinities. Hence, if
in forming the function Z(x), = 02(x)/01(x), the places A, A' have been
disregarded, so that the polynomials 0l(x}, 02(x) do not vanish in these
places, the function Z (x) will take the same value at A as at A', and
we shall obtain
yx»ai + ...... _j_ y*^ _ 0
Hence we obtain the result : if, in the formation of the integrals of the
first kind for a given fundamental curve, we overlook the existence of a
certain number, say B, of double points, we shall obtain p + 8 integrals, where
p is the true deficiency of the curve; and these integrals will be linear
aggregates of the actual integrals of the first kind and of 8 integrals of the
third kind. If in the formation of the rational functions also we overlook
the existence of these double points, Abel's theorem will have the same form
of equation for the p + 8 integrals as if they were integrals of the first kind
(cf. §§ 83, 90, and Abel, (Euvres Camp., Christiania, 1881, Vol. I. p. 167).
For example, let a1? ..., aQ be arbitrary places in which r + 1 ^-poly
nomials vanish (Chap. VI. §§101, 93). Take q(=Q-p + r+l) arbitrary
157J DISAPPEARS FROM THE EQUATION. 217
places d, ..., cq, and so determine the set d, . .. , CQ coresidual with alt ..., «y.
A rational function, f(#), which has the places al} ..., aQ for poles and the
places d, ..., CQ for zeros is quite determinate save for a constant multiplier.
Let j,\, . . . , XQ be any set of places at which f (x) has the same value, A say,
so that #!, ... , XQ are the zeros of £(#) — .4 ; then, as a1} ... , ac are the poles
of £ (#) — A . we have
p*. , <*, p,u, «,, , £(Ci)-VJ
^,,2-f ^r,,,, 10g£(Cs)_^>
and as f(d) = £(c.,) = 0, the right hand is zero.
Hence, calling the places where a definite rational function has the same
value a set of level points for the function, we can make the statement — the
level points of a definite function satisfy the equations
c1} c., being any two of the zeros of the function.
In particular, when q = l, the sets of level points are the most general
sets coresidual with the poles or zeros of the function. Hence, if xly ...,xp+1
be any set of places coresidual with a fixed set c,, ca, ...,cp+li in which no
^-polynomials vanish, we have the equations
157. Ex. i. We give an example of the application of Abel's theorem.
For the surface associated with the equation
the integral
f.rP4-/?-/rP-l_l- _j_c
dx
y
is of the second kind, becoming infinite only at the (single) place # = oo. Consider the
rational function
which, for general values of A,..., L0, is of the (2jo + l)th order, its zeros, for instance,
being given by
To evaluate the expression
(- 1
\dt Z-
the place .* = « l^ing the only one to be considered, we put x = t~'> and obtain
218 EXAMPLES. [157
~-
dl ^P^fip-aT ...... _2
dt~ 2 ~*'
and therefore
dl 1 1 1 • , A — A0 1 cl
wherein the coefficient of t~l is ^ (A -^ A0) (1— /i)~3.
Hence, if .1^, ..., ^'2,, + l be the zeros, and «!,..., «2P + 1 be the poles of Z, we have
Now the zeros of Z are zeros of the polynomial
denoting the values of y by yl, ... , y.iv + v and using F(x) for (x~x^ ...... (^'-^p + i),
where (xlty^,..., (xp + l,yp + l) are any p + l of the places (x^y^..., (xtj>+1, y2, + 1), we
have, from the jt? + 1 equations
and hence, if 6^ 62, ... be the values of y when .v=a1, a2, ..., and ^T0(^) = (^ — «x) ...
(^-ap + 1), we have
7^'^+ ...... +/«*H-i'ffl^i=i 2 Fr 2 F7^)-
i=l * ^*V i = l 0 \M't''
If in the integral / the term .vp be absent, the value obtained for the sura
1*1 '«l+ ...... + 7*8^-1 ' "SH-I
will be zero.
The reader will notice that for p = l, we obtain an equation from which the equation
can be deduced, ult u%, u3 being arguments whose sum is zero ; and that the algebraic-
equation whose roots are x1}..., x2p + 1 gives
/p + l y. \2
#„ + ...... +^ + i = J^2 = i 2
which for = \ becomes
Io7] HYPKRELLIPTIC CASE. 219
Ex. ii. If Y) Z be any two rational functions, and u any integral of the first kind,
prove by the theorem
/ 1
\(Y-b}(Z
du dx
~
-J) dx
that the sum of the values of (Y—b)~l dujdZ, at all places where £=c, added to the sum
of the valuer of (Z— c)~l du/dY &i all places where F=6, is zero.
It is assumed that all the zeros of the functions Y- b, Z—c are of the hrst order.
Hence prove the equation
2 T— = 5 (du Z(x}-n
8
where a1( ..., ag are the places at which Z(x) = v, x1, ..., XQ the places at which Z(x)=p,
and the suffix on the right hand indicates that the values of the expression in the brackets
are to be taken for the n places of the surface at which x=b.
It is assumed that there are no branch places for x=b.
Ex. iii. If </> (x) be any integral polynomial in .r, y* = (x, l)2p + 2> =/(•*•') say. and M (x),
N(x] be any two integral polynomials in x of which some coefficients are variable, and
f(x) . M^(x}-^(x) = K(x-xl} ...... (x-xgl
where K is a constant or an integral polynomial whose coefficients do not depend upon
the variable coefficients in M(x], N (x\ and yi,...,ys be determined by the equations
yiM(Xi) + N(xi) = 0, then, on the hypothesis that s is not one of the quantities xlt ..., XQ>
and is not a root of/(#) = 0, prove that
(*
J
-
where C is a constant, and R is the coefficient of - in the development of the function
) ! N
*
in descending powers of x ; herein the signs of Jf(x) , */f(zj are arbitrary, but must be
used consistently.
Shew that the statement remains valid when f (x) is of order 2p + l (in which case the
development from which r is chosen is to be regarded as a development in powers of */x) •
prove that r is zero when <£ (a?) is of order p, or of less order. Obtain the corresponding
theorem when 2 is a root of f(x) = 0.
Ex. iv. The result of Ex. iii. is given by Abel ((Euvres Compl., Vol. i. p. 445), with a
direct proof. We explain now the nature of this proof, in the general case. Let/ (y, x) = 0
be the fundamental equation, and let 6 (y, x) be a polynomial of which some of the
coefficients are variable ; if y^ ... , yn be the n conjugate roots of / (y, x) = 0 corresponding
to any general value of .r, the equation
r (x) = e (#!, .») 6 (y2, x} ...... B (yu, .»:) = 0,
gives the values of x at the finite zeros of the polynomial 6 (y, x). Suppose that the
left-hand side breaks into two factors F0 (.v) and F (x), of which the former does not
contain any of the variable coefficients of 6 (y, x}. Let £ be a root of F(x) = 0, and
Vu ••-, in be the corresponding values of ij ; then one or more of the places (£, ^j), ...... ,
220 EXPLANATION OF ABEI/S PROOF. [157
(£> »7n) are zeros of 6 (y, x) ; fix attention upon one of these, and denote it by (£, rj). Then
if, by a slight change in the variable coefficients of 6 (y, x\ whereby it becomes changed
into 6 (y, x) + 80 (y, x\ F (x) become F (x) + 8F O), the symbol 8 referring only to the
coefficients of 6 (y, x\ and £ become £ + o?£, we have the equations
" (£)<*£=<>,
8,- (|)= 20 Oh, £) ...... 5 fo.,, £) e (,i + 1, |) ...... <9 (,n, ^) 80 fa, £),
i = l
where /*' (g) = dF((;)/dg. Denote now by U (x) the rational function of x, given by
U(x) = 2 d (y1? x) ...... 6 (#_!, #) 0 (yi + 1, a,-) ...... 0 (yn, x) W (y4, *•);
»=i
then if /i (a-', y) be any rational function of x and y, we have
where, on account of ^ (r/, ^) = 0 we can write
and
= 0 (I), say,
0 (£) being a rational function of £ only. Taking the sum of the equations of this form,
for all the zeros of 6 (y, x}, we have
herein the summation on the right hand can be carried out, and the result written as the
perfect differential of a function of the variable coefficients of 6 (y, .r), in fact in the form
(#, y) * log
as we have shewn.
For example, when
/ (y, x) =y3 + x3 - 3ayx -I, 6 (y, x) =y - mx - n, we have F0 (x) — 1,
F (x} = x3 + (mx + n)3 — Sax (mx + n) — 1 ,
and
&<% _ _ 3£,8f (|)_ _ _ 3^rW(ga«i + a>Q _ _ 3g(mg + ?Q(gam + an) >H|) (
7s -«l /T^^'Tl)" /'(«7)^«) *"(£) ^(f)'
Now ^^)_
-
, , „ cnwc xw (x) 3xm8m\ „. /«l»— a\
and hence 2 -— — —„ = „, / + ^- , = - 38 ^ 5 ,
•7 a^ L*-W l+m3Jx=cc \l+m3/
as is easily seen. From this we infer
n — a fmn - cA ^ A'J - ^2
2 I .-^^.= -3
» =
157] ABEL'S FORM OF THE RESULT. 221
In this example it is easily seen that the integral is only infinite when x is
infinite; putting x^t~\ the equation f (y, #) = 0 gives y= -a>t~ l-au* + At + Bt* + ,
where « = 1, or (-l±V^3)/2; then log 6 (y, x) dl/dt, =\og (y - m,r - n) \xyl(yz-ax)}
dxjdt, has (a»*+n) a>2/(« + m) for coefficient of t~\ and we easily find
a+n a> 2 a> _
m + I TO + O) & m + o)2"
.Ek. v. If Y, Z denote any two rational functions (in x and y\ such that there is no
finite value of x for which both have infinities, and 2 (YZ) denote the sum of the n
conjugate values of YZ for any value of x, and [2 (YZ}\x_ay, denote the sum of the
coefficients of (x- a)'1 in the expansions of the rational function of x, 2 (YZ), for all finite
values of x for which Y is infinite, and [S (YZ)^ denote the coefficient of x~l in the
expansion of 2 (YZ) in descending powers of x, it is easy (cf. § 162 below) to prove that
wherein, on the left hand, the dash indicates that the sum is to be taken only for the
finite places at which Z is infinite. Hence if 7 be any Abelian integral, = \R(x,y) d.v,
we have
S log I (a ,))^-[. (£« log 0 (y, •
Hence, if we assume that 0 (y, x) has no variable zeros at infinity, we can obtain
Abel's theorem in the form
wherein the summation on the left refers to all the zeros of 6 (y, x).
This is the form in which the result is given by Abel ((Euvres Compl., Christiania, 1881,
Vol. i. p. 159, and notes, Vol. ii. p. 296), the right hand being obtained by actual
evaluation of the summation which we have written, in the last example, in the form
_, ^M_
* (&**(&'
The reader is recommended to study Abel's paper*, which, beside the theorem above,
contains two important enquiries ; first, as to the form necessary for the rational function
dl/dx, in order that the right-hand side of the equation of Abel's theorem may reduce to a
constant, next, as to the least number of the integrals in the equation of Abel's theorem,
of which the upper limits may not be taken arbitrarily but must be taken as functions
of the other upper limits. Though the results have been incorporated in the theory here
given (§§ 156, 151, 95), Abel's investigation must ever have the deepest interest.
K.I: vi. Obtain the result of Ex. i. (§ 157) by the method explained in Ex. iv.
* Which was presented to the Academy of Sciences of Paris in Oct. 1826, and published by
the Academy in 1841 (Mi'moirc* par dirfrx xavants, t. vii.). During this period many papers were
published in Crelle's Journal on Abel's theorem, by Abel, Minding, Jiirgensen, Broch, Richelot,
Jacob! and Rosenhain. (See Crelle, i — xxx. I have not examined all these papers with care.
Jiirgensen uses a method of fractional differentiation.)
222 CONVERSE OF ABEL'S THEOREM. [157
Ex. vii. Prove that the sum of the values of the expression
U.v
J '
wherein v is any linear expression in the homogeneous coordinates x, y, z, U is any
integral polynomial of degree m + n — 3, J is the Jacohian of any two curves /=0, 0 = 0,
of degrees n and m, and the line v = 0, and the sum extends to all the common points
of/=0 and 0 = 0, vanishes, multiple points of/=0, 0 = 0 being disregarded.
Hence deduce Abel's theorem for integrals of the first kind.
(See Harnack, Alg. Diff. Math. Annal. t. ix. ; Cay ley, Amer. Journ. Vol. v. p. 158 ;
Jacobi, theoremata nova algebraica, Crelle, t. xiv. The theorem is due to Jacobi ; for
geometrical applications, see also Humbert, Liouville's Journal (1885) Ser. iv. t. i. p. 347)*.
Ex. viii. For the surface
ya=tf> (*)*(*), =/(*),
wherein 0 (#), ^ (x] are cubic polynomials in #, prove the equation
wherein .vlt #2, | and ml5 m2, y are coresidual with the roots of 0 (#)=0, and |, y are the
places conjugate to £ and y ; conjugate places being those for which the values of x are
the same.
158. When the places xl)...,xq are determined as coresidual with
the fixed places an ... , aq, p — r — l of the places xl} ...,xq are fixed by
the assignation of the others. Hence the p + 1 relations, Avhich are given by
Abel's theorem,
cannot be independent. We prove now first of all that the last may
be regarded as a consequence of the other p equations. In fact, if x\, ... , Xq
and alt ..., aqbe any two sets of places, sucJi that, for any paths of integration ,
(i = l,2, ...,p), wherein u*\a, ...,u*'a are any set of linearly independent
integrals of the first kind, &>,-(1 , ... , a>'i>p are the periods of the integral u{' , and
M-i, ..., M'p are rational integers independent of i, then there exists a rational
function having the places a^, ..., aq for poles and the places ti\, ..., Xq for
zeros.
For if vXi'a, ..., vl'n be the normal integrals of the first kind, so that we
have equations of the form,
x, a f-i x, a ,-i x, a
Vi = C'j( j Wj + ...... + Ci<p Up ,
* Further algebraical consideration of Abel's theorem may be found in Clebscb-Lindemann-
Benoist, Lecona sur la Geometrie (Paris 1883) Vol. iii. Geometrical applications are given by
Humbert, Liouville's Journal, 1887, 1889, 1890 (Ser. iv. t. iii. v. vi.).
158] THE INDEPENDENT AND SUFFICIENT EQUATION. 223
wherein C^, ..., Ciip are constants, and therefore, also,
Cit j ta^j + ...... + Gi> p copi j = 0 or 1, according as i ^=j, or i =j,
and
@i, I m'\,j + ...... + Ci,P(0'p,j = ri,}>
we can deduce
v?'ai + ...... +v?'aa = Mt + Ml'Titt + ...... +M'pTitp.
Consider now the function
«// x Uxa + ...... + nx°a -2"i(M\vx'c+ ...... +M'vx'°)
Z(x) = e " ' xQ>aa PP '
c being an arbitrary place.
Herein an integral, Tix\,a,> suffers an increment 2jri when a; makes a
circuit about the place ^ ; but this does not alter the value of Z (#). And
in fact Z(x) is a single-valued function of x; for the functions 11% aa have
no periods at the first p period loops, while, if x describe a circuit equivalent
to crossing the t-th period loop of the second kind, the function Z(x) is only
multiplied by the factor
or eiwlMi, whose value is unity.
Further the function Z (x) has no essential singularities; for it has poles
at the places alt ... , ay, and is elsewhere finite.
Since the function has zeros at xl , . . . , XQ and not elsewhere, the state
ment made above is justified.
Ex. i. It is impossible to find two places y, £ such that each of the p integrals wf- f is
zero. For then there would exist a rational function, given by
having only one pole, at the place y. (Of. § 6, Chap. I.) It is also impossible that the
equations
wherein 3flt ..., Mp, M\, ..., M'v are rational integers independent of i, should be
simultaneously true.
Ex. ii. If p equations, of the form
exist, y, and y2 are the poles of a rational function of the .second order, and the surface
hyi>erelliptic. (Chap. V. § 52.)
224 THE NUMBER OF INDEPENDENT EQUATIONS. [159
159. In regard now to the equations
11.
which express that the places a?1} ...,XQ are coresidual with the places
alt . . . , O,Q, if r + 1 be the number of (^-polynomials which vanish in the places
al} ...,aQ (Chap. VI. § 93), or (Chap. III. §§ 27, 37) the number of linearly
independent linear aggregates of the form
wherein C1,...,GP are constants, which vanish in these places, then,
Q — p + r + 1 of the places xl,...,Xq can be assumed arbitrarily, and the
equations are therefore equivalent to only p — r — 1 equations, determining
the other places of xl , . . . , XQ in terms of those assumed. This can be stated
also in another way : the p differential equations
express that the places oc1, ...,XQ are coresidual with the places #j + dx^ , . . . ,
XQ + dxq ; if the places xly ...,XQ have quite general positions these equations
are independent ; if however T + 1 linearly independent linear aggregates, of
the form,
~ du^ ~ dup
Cld^c + ...... +^^-°'
wherein Clt ...,CP are constants, vanish in the places xlt ...,Xy, then the p
differential equations are linearly determinable from p — r — 1 of them.
Ex. i. A rational function having .r1} ..., .% as poles of the first order, and such that
Xu ..., Xp are the coefficients of the inverses of the infinitesimals in the expansion of
the function in the neighbourhood of these places, can be written in the form
\ r-*' f > T>X" c •
— AT 1 — ...... — \o 1 ;
i /r f
X, Xy
the conditions that the periods be zero are then the p equations
A1Qi(#i) + ...... +XBOi(A'B)=0, (i=l, 2, ...,p).
But, if we take consecutive places coresidual with xl, ..., .ry, and tl,...,t(l be the
corresponding values of the infinitesimals at .vlt ..., ,re, we also have
thus, if the first q ( = Q — f> + r + 1) of tlt ..., tu be taken proportional to X,, ..., X,, we shall
have the equations
'iAi= ...... =*«/*«•
Ex.\\. When the set .r1,...,x(!, beside being coresidual with o^, ...,afi, has other
specialities of position, Abel's theorem may be incompetent to express them. For instance,
in the case of a Riemann surface whose equation represents a plane quartic curve with
two double points, there is one finite integral ; if al9 ..., «4 represent any 4 rollinear points,
and .r1? ...,.r4 represent any other 4 collinear points, the equation of Abel's theorem is
1GO] ABEL'S DIFFERENTIAL EQUATIONS. 225
but this equation does not express the two relations which are necessary to ensure that
xlt ...,%i are collinear ; it expresses only that #l} x2, x3, x± are on a conic, S, passing
through the double points, or that x^, .v2, xa, x± are the zeros, and ax, ...,«4 are the poles
of the rational function S/LL0, where L=Q is the line containing aly...,at and L0=Q is
the line joining the double points.
100. From these results there follows the interesting conclusion that
the p simultaneous differential equations
have algebraical integrals, Q being > p, and «,, ...,up being a set of p linearly
independent integrals of the first kind. The problem of determining these
integrals consists only in the expression of the fact that #,, ..., ary con
stitute a set belonging to a lot of coresidual sets of places.
The most general lot will consist of the sets coresidual with Q arbitrary
fixed places a,, ...,aQ, in which no (^-polynomials vanish. But the lot does
not therefore depend on Q arbitrary constants; for in place of the set
a,, .... nQ we can equally well use a set Alt ..., AQ, whereof q, =Q-pt places
have positions arbitrarily assigned beforehand ; in other words, all possible
lots of sets of Q places with multiplicity q can be regarded as derived from
fundamental sets of Q places in which q places are the same for all. A lot
depends therefore on Q-q,=p} arbitrary constants, and this number of
arbitrary constants should appear in the integrals of the equations (Chap VI
§96).
We may denote the Q arbitrary places, with which xlt...,x(i are coresidual,
by A1,...,Aq,al,...,ap, so that A1,...,Aq are arbitrarily assigned before
hand, in any way that is convenient, and the positions of «,, ...,ap are the
arbitrary constants of the integration.
Then one way in which we can express the integrals of the equations is
as follows: form the rational function with poles, of the first order, in the
places #1, ..., xQi and determine the ratios of the q + 1 homogeneous arbitrary
coefficients entering therein, so that the function vanishes in A^^^Aq.
Then the function is determined save for an arbitrary multiplier, and
must vanish also in a,, ...,ap. The expression of the fact that it does so
gives p equations, each containing one of ctj, ..., ap as an arbitrary constant.
From these p equations we may suppose p of the places aslt ...,XQ, say
*i,..., «P, to be expressed in terms of o^....^ and xp+l, ...,XQ (and
Al,...,Aq). The resulting equations may be derived also by forming the
general rational function with its poles in a,,..., ap> A, ,..., Aq and eliminating
the arbitrary constants by the condition that this function vanishes in
xii ®p+i, tfp+2, ...,acQ,i being in turn taken equal to 1, 2, . . . , p.
B- 15
226
EXAMPLES OF
[160
For example, for Q=p + 1, if ty (as, a; z, cl5 ..., cp) denote the definite
rational function which has poles of the first order in the places z,cl,...,cp,
the coefficient of the inverse of the infinitesimal at the place z being
taken = — 1, which function also vanishes at the place a (Chap. VII. § 122),
then a complete set of integrals is given by
; #p+1, #!,..., #p) = 0 = ...... = ^(ap,A; xp+lt x1} ..., xp\
and a complete set is also given by
^ (®I,XP+I', A, alt ..., ap) = 0 = ...... = i/r (xp, xp+1 ; A, a,,..., ap).
The first of these integrals is in fact the equation
dup dup
dxl ' dxz '
dP dP
dxri
du
dx
p+l
dP
= 0,
wherein P = P^'^, and may be regarded as derived by elimination of
dx1} ..., dxp+l from the p given differential equations and the differential of
the equation (§ 156)
, and (J., tt:, ..., <%,) are coresidual
- a,, A
which holds when (a?1} ..., ^+1), (c^
sets.
^r. i. For p = l, the fundamental equation being yi = (x^ 1)4=X2.^4 + ..., shew that the
differential equation
ofei c^2 = 0
y\ yz
has the integral
where 62 = (a, 1)4. (Here the place A has been taken at infinity.)
Shew also that this integral expresses that the places (#1} T/J), (.r2, y2)> (#> — &)> are the
variable zeros of the polynomial — y+jo + ^-X^2, when p and <? are varied.
Ex. ii. For jo = 2, the fundamental equation being 3/2 = (#, l)c = X2^i6 + ..., using the
form of the function ty (xi a> z> ci> •••> CP) given in Ex. ii. § 132, Chap. VII., and putting
the place A at infinity, obtain, for the differential equations
the integral
\Ai3C-t \Aj3Cn \AtJ(j o _.
_J+ _2 + _3 = 0j
2/1 3/2
3/i
<i z^ _
— T , — "«
3/2 3/3
3/3
to - a) F' to) to - a) ^' (*a) (^3 - «) ^" W ^ (a)
wherein /ri(.r) = (^-^1)(.'r-,'r.j) (.r — .rs), b2=(a, l)c, and the position of the place (a, 6) is
l(jl]
ABEL'S DIFFERENTIAL EQUATIONS.
227
the arbitrary constant of integration. By taking three positions of (a, 6) we obtain a
system of complete integrals.
Shew that this integral is obtained by eliminating pt q, r from the equations which
express that the places (xlty^)y Cr2J #2)1 (xa> 2/3)1 (a> ^) are zeros of the polynomial
—y — \y? +px2 + qx+r.
Ex. iii. For the case (p = 3) in which the fundamental equation is of the form
(x, y)4 being a homogeneous polynomial of the fourth degree with general coefficients, etc.,
prove that an integral of the equations
is given by
where
(2, 3, 4) ^ + (3, 1, 4) Uz + (l, 2, 4) ^3-(l, 2, 3) ^4 = 0,
(2, 3, 4) =
:r2 xz ,r4
3^2 3/3 2/4
1 1 1
etc.,
and
/(6, a) being =0, and the position of (a, 6) being the arbitrary constant of integration.
A complete system of integrals is obtained by giving (a, 6) any three arbitrary positions.
To obtain these equations the place A has been put at x=0, y = Q.
Ex. iv. When the fundamental equation is «4+2/4 = l, shew, putting the place A at
a?=l, ?/ = 0, that, as in Ex. iii., we have integrals of the form
(2, 3, 4) CT1 + (3, 1, 4) U2 + (l, 2, 4) U3-(l, 2, 3) ^ = 0,
wherein
and
„ _i
161. The method of forming the integrals of the differential equations
which is explained in the last article may also be stated thus: take any
adjoint polynomial ty which vanishes in the Q places Alt ..., Aq, aly ..., ap;
let Oi, ..., CK be the other zeros* of i/r; let the general adjoint polynomial
of the same grade as t/r, which vanishes in C,, ..., CR, be denoted by
X, Xj, ...,\9 being arbitrary constants. By expressing that the places
xi> xp+i, #p+2> •••, %Q are zeros of this polynomial we obtain a relation
whereby #; is determined from xp+l, ..., XQ in terms of the arbitrary positions
Beside those where/' (y) or F' (77) vanishes (cf. Chap. VI. § 86).
15—2
228
METHODS OF EXPRESSING
[161
a1} ..., ap (and Alt ..., Aq). By taking i = 1, 2, ..., p we obtain a complete
system* of integrals.
Now instead of regarding the set Alt ..., Aq, al} ..., ap as the arbitrary
quantities of the integration, we may regard the set C1,...,CK as the
arbitrary quantities, or, more accurately, we may regard the p quantities
upon which the lot of sets coresidual with Clt ..., CR depends, as the
arbitrary quantities. To this end, and under the hypothesis that no
^-polynomials vanish in the places Clt..., CR, imagine a set of places
Bl,...,BB-p, bl}...,bp determined coresidual with G^,...,CR> in which
BI,..., BR-P have any convenient positions assigned beforehand, so that the
lot of sets coresidual with C1 , . . . , GR depends upon the positions of 61,..., bp.
Let a general adjoint polynomial with Q + R variable zeros be of the form
wherein fj,, ..., ^ are arbitrary constants, and k is for shortness written for
Q + R — p. Then an integral of the differential equations under con
sideration is obtained by expressing that the places
JJl, ..., £>R—p, Vif ..., Op, OCi, Xp+}, Xp+i, ••', %Q
are zeros of the polynomial ® ; and a complete system of integrals is
obtained by putting i in turn equal to 1, 2, ..., p.
Similarly a complete set of integrals is obtained by expressing that
the places
SS1} ..., Xp , (Kp+i , . . . , Xq , GI, i>i , . . . , JJR—p
are zeros of the polynomial @, i being taken in turn equal to 1, 2, ..., p.
In this enunciation there is no restriction as to the value of R, save that
it must not be less than p.
Ex. i. For the general surface of the form
/ (y, x} - (#, y\ + (x, y)3 + (x, y)z + (.r, y\ + constant = 0,
a set of integrals of the equations
tlcAj _
f/TyT0'
4
?/(y<)
T = °.
is given by
y\
y
^2 ^15
A B I
* And we can of course obtain quite similarly a set of p integrals, each connecting
xl , . . . , xg , Alt ..., Av, and one of the arbitrary positions a, , . . . , a,> .
1C1] THE SOLUTION OF THE EQUATIONS. 229
where /((!>;, «*)=(), f (B, J) = 0, i=l, 2, 3, mid the place (.1, B) may be taken at any
convenient position.
Ex. ii. Taking as before (J =/» + !, and considering the hyperelliptic case, the funda
mental equation being
we require a polynomial having R -\-p + 1 variable zeros : such an one is
Q=-y + X.u"
It being equal to p, and we have
where /* (a-) = (# - .^ ...... (*--^f) + 1), 0 (.v) = (je-bl) ...... (x-bp).
An integral of the differential equations may be obtained by eliminating F, 6-',..., H
from the equations expressing that the places
0|, ..., Op, #f, #p + 1
are zeros of the polynomial e, or from the equations expressing that
•vn ••••> -Vp) **;:> + 1> "»
are zeros of this polynomial, and a complete system of integrals, in either case, by taking
i in turn equal to 1, 2, ..., p.
Or a complete system of p integrals may be obtained by eliminating F, G, ..., H from
the 2p + 1 equations obtained by equating the coefficients of the same powers of x on the
two sides of the equation.
We may of course also take e in the form
then R=p + l, and the places Bl, ..., BR_V are not evanescent ; putting the place Bl at
infinity we obtain E=\, as above.
Ex. iii. The integration in the previous example may be carried out in various ways.
By introducing again a set of fixed places «u ..., ap, A, coresidual with x^ ..., xp, xp + l,
we can draw a particular inference as to the forms of the coefficients F, G, ..., H. For if
U (x) denote X.r" + l + Fx* + ... + G, and U0 (x) denote what U (x) becomes when x1,...,xp + l
take the positions alt ..., op, A, the coefficients F, G, ..., H being then F0) G0, ..., ff0,
and also F0(x) = (x-a1) ...... (x - a}>} (x - A ), then, because each of the polynomials
-y+ U (x\ -y+U(}(x) vanishes in the places 61}..., bp, the polynomial U(x)-U0 (x}
must divide by 0 (#), namely U (x} = 170 (x) + 1 <f> (#), where t is a variable parameter;
or, if we write 0 (x) = x» + t1x*>-i + ...... + tp, tlt...,tp being then regarded, instead of
&!,..., 6P, as the arbitrary constants of the integration, we have
and the quantities G-^ F, ..., Jf-t,, F are constants in the integration, being unaltered
when the places ^, ..., xp + l come to alt ..., av, A. Hence we can formulate the following
result: let the ^ + 1 quantities F9, G0, ..., ff0 be determined so that the polynomial
-y+U0 (x) vanishes in the fixed places at, ..., ap, A. Then denoting (^-^...(^-ap)
(x-A) by F0(x), the fraction
is an integral polynomial; denote it by (p-2F9 X) (jev + tl x»~i+ ...... + <„), so that
230 THE HYPERELLIPTIC CASE. [161
<j>0,tl,...ytp are uniquely determined in terms of the places a,, ..., ap, A, and put
F(x) for x» + t1xp~1 + ...... + tp. Then xlt ..., xp + l are the roots of the equation
]}*==(»-2F» X) F, (*)-* U0 (*)
and the set x1}...txp + l varies with the value of t, which is the only variable quantity
in this equation. By equating the coefficients of the various powers of x in the
polynomial on the left-hand side of this equation to the coefficients in the polynomial
(n-2F0\) F(x\ we can express each of the symmetric functions
MI = X^ T ...... T Xp + i
as rational quadratic functions of a variable parameter t, containing definite rational
functions of the variables at the places a15 ..., ap, A ; the place A may be given any
fixed position that is convenient ; the positions of the places a1? ..., ap are the arbitrary
constants of the integration.
Ex. iv. By eliminating t between the p + l equations obtained at the end of Ex. iii.
we obtain the complete system of p integrals. In particular any two of the quantities
/*!, /J2, ... are connected by a quadratic relation, and any three of them are connected by
a linear relation (Jacobi, Crelle, t. 32, p. 220).
Ex. v. From the equation
we infer
where hl—x1 + ...+xp + l ; hence if a be the value of x at a branch place of the surface,
we have from Ex. ii.
and if, herein, a be put in turn at any p of the branch places of the surface, the resulting
values of <f> (a) may be regarded as the arbitrary constants of the integration, and the
resulting equations as a complete set of integrals ; and if X = 0, as we may always suppose
without loss of generality (Chap. V.), we thus obtain the p integrals
Clt ..., Cp being the constants of integration (Richelot, Crelle, xxiii. (1842), p. 369. In this
paper is also shewn how to obtain integrals by extension of Lagrange's method for the
case p = I. See Lagrange, Theory of Functions, Chap. II., and Cayley, Elliptic Functions,
1876, p. 337).
Ex. vi. By comparing coefficients of x2p in the equation of Ex. ii., we obtain
v - (2X0 + F*} = 0* -
where h1=x1 + ... + xll + 1; hence prove that
r-l
162] ABEL'S THEOREM FOR A CURVE IN SPACE. 231
by Ex. ii. the right-hand side is a constant in the integration ; hence this equation is an
integral of the differential equations; in particular if X = 0, fi = 4, which is not a loss of
generality, we have the integral
where C is a constant ; this is a generalization of the equation, for p — 1,
(cf. Ex. i. § 157).
Ex. vii. Shew that if the fundamental equation be
then another integral is
,.,..,Vl p«_* -i'_£ (i+...+ _L)_ „ (L+...+ j_y=Coilst.
Lr=i^* wJ Vi XP+J Vft *+i/
(Richelot, loc. cit.)
Ex. viii. If a0, 0^ be the values of x at two branch places of the surface, obtain the
equations
(Of-^) ...... (0, -Op) (oo-J) ...... (Oo-Op)
wherein the quantities .4, ...,ap are the values of ^ at fixed places coresidual with
*'i,..., *'P+I> Pi ^8 an absolute constant, and ^ is a parameter varying with the places
#!,..., Xp+i- Take i in turn equal to 1, 2, ..., (p + 1), and, eliminating /x, we obtain a
complete set of integrals. In particular if the left-hand side of this equation be denoted
by GI we have such equations as
(Gi - 1) PJ pk (PJ - pk) + (Gj - 1) Pk Pi (PA - Pi) + (Gk - 1) pi PJ (pi - PJ) =0.
(Weierstrass, Collected Works, Vol. I. p. 267.)
162. The proof of Abel's theorem which has been given in this chapter
can be extended to the case of an algebraical curve in space. Taking the
case of three dimensions, and denoting the coordinates by a, y, z, we shall
assume that for any finite value of x, say x = a, the curve is completely given
by a series of equations of the form
x = a + ,
y = Pl(tl) , y = P2(t2) , ......... ,y = P*(fe) , (D)
wherein wl + I, ..., wk + I are positive integers, ^,...,^ are infinitesimals,
and PI, Q1} ..., Pk) Qk, denote power series of integral powers of the variable,
with only a finite number of negative powers, which have a finite radius
of convergence. The values represented by any of these k columns, for all
values of the infinitesimal within the radius of convergence involved, are the
coordinates of all points of the curve which lie within the neighbourhood
of a single place (cf. § 3, Chap. I.) ; the sum
232 PROOF ENTIRELY ANALOGOUS [162
is the same for all values of x, and equal to n, the order of the curve. A
similar result holds for infinite values of x ; we have only to write - for x — a.
x
We assume further that any rational symmetric function of the n sets
of values for the pair (y, z), which are represented by the equations (D), is a
rational function of x.
Then we can prove that if R (x, y, z) be any rational function of x, y, z,
dx
the sum of the coefficients of t~l in the expression R (x, y, z) -j. , at all the
k places of the curve represented by the equations (D), is equal to the
coefficient of in the rational function of x,
x — a
5 (x, ylt zd + R(x,y2, z2) + + R (x, yn, zn}.
And further that the sum of the coefficients of t~l in R (x, y, z) -^ at all
the places arising for x = oo is equal to the coefficient of — in the expansion
of the same rational function of x, namely, equal to the coefficient of t'1 in
U (x) -r , when x = -.
d/t t
Hence, the theorem
which holds for any rational function, U (x), of a single variable (as may be
immediately proved by expressing the function in partial fractions in the
ordinary way), enables us to infer, in the case of the curve considered, that
also
By this theorem, applied to the case
T ]_ d^ R ^ . dx
\_R (x, y, z) dx '
we can prove that the number of poles of R (x, y, z) is equal to the number
of its zeros, and therefore also equal to the number of places where R (x, y, z)
has any assigned value /*,, a place being counted as r coincident zeros when
the expression, in R (x, y, z), of the appropriate values for x, y, z, in terms
of the infinitesimal, leads to a series in which the lowest power of t is tr ;
similarly for the poles.
162] TO THAT FOR SIMPLER CASE. 233
Hence, if / be any integral of the form j R (x, y, z) dx, we can apply
this theorem in the form
dt Z-
Z being any rational function of a?, y, z, and so obtain, as before (§§ 154, 155),
the theorem
and if Z is of the form 02 (x, y, z)/0i (a, y, z}, where B2, @i are integral poly
nomials, we can put the right-hand side
= log ' J'
[dt 8ei(x,1J,
wherein xl , . . . , xk are the places at which Z = 0, or 02 (x, y, z) = 0, and
ttj , . . . , ak are the places where Z—<x> or 01 (x, y, z) = 0, and the places
to be considered on the right hand are the infinities of dl/dt.
The reader may also consult the investigation given by Forsyth, Phil. Trans., 1883,
Part i. p. 337.
Take for example the curve which is the complete intersection of the cylinders
For any finite value of .v, except x=0 or x= 1, we have 4 places given by
y= ±*Jx (1— &•), z= ±\/x.
For infinite values of.*;, putting x = -„ we have two places given by
. 1 1
For x=l, putting x=l + f2, we have two places given by
& = & + ... , y =
z=+(l+W+t..), a=
For .£ = 0, putting .v = f2, we have two places given by
*-*(!-*«•-...) I yy=-<(l_i^_ )?
«=< , z=t
and, at j,' = 0,y=0, 2 = 0, d.v : dy : dz = -2t : 1 : 1 or =2« :-l : 1=0 : 1 : 1 or =0 :- 1 : 1
so that there is a double point with x = Q,y=±z for tangents.
Consider now if— , from the intersections of z + ax + by~0 to those of s + a'x + b'y = 0.
234 EXAMPLE. [162
Put /= / — ; then -=- , = -- j- , when x is near to 0, has, for one value,
J yz at' yzdt
. -W ...) . 1+6'
whlle lo ° Lr108 T+6
+
and the contribution to the sum ( -=- log — T I , is 2 log ---
\dt 6 z+ax + byjt~l & 1 + 6
If we take the other place at x=Q we shall get, as the contribution to
dt
the quantity — 2 log - — j- .
Thus, on the whole we get, at #=0,
It is similarly seen that no contribution arises at the places #=1, x = oo.
Thus on the whole
[ <**! I f
J ^ 7T J
Now from the equations zl-\-axJ-\-by1 = 0, z2 + ax.2-\-by2 — ^ we find
b =
and thus
— ==-Jlqg
which is a result that can be directly verified.
168] 235
CHAPTER IX.
JACOBI'S INVERSION PROBLEM.
163. IT is known what advance was made in the theory of elliptic
functions by the adoption of the idea, of Abel and Jacobi, that the value
of the integral of the first kind should be taken as independent variable, the
variables, x and y, belonging to the upper limit of this integral being regarded
as dependent. The question naturally arises whether it may not be equally
advantageous, if possible, to introduce a similar change of independent
variable in the higher cases. We have seen in the previous chapter that, if
u* l a , . . . , ux' a be any p linearly independent integrals of the first kind, the
p equations
justify us in regarding the places x1} ..., xp as rationally determinable from
the arbitrary places a^, ..., aQ, xp+1, ..., x^ hence is suggested the problem,
known as Jacobi's inversion problem*, which may be stated thus: if
Ul} ..., Up be arbitrary quantities, regarded as variable, and ai} ..., ap be
arbitrary fixed places, required to determine the nature and the expression of
the dependence of the places xlt ..., xp, which satisfy the p equations
upon Hie quantities U1} ..., Up. It is understood that the path of integration
from ar to xr is to be taken the same in each of the p equations, and is not
restricted from crossing the period loops.
164. It is obvious first of all that if for any set of values U1} ..., Uv
there be one set of corresponding places xlt ..., xp of such general positions
that no ^-polynomial (§ 101) vanishes in them, there cannot be another set
of places, Xi, ... , Xp, belonging to the same values of Ult ... , Up. For then
we should have
* Jacobi, Crelle xin. (1835), p. 55.
236 GENERAL EXPLANATION [104
and therefore (§ 158, Chap. VIII.) there would exist a rational function
having xl} . .., xp as poles and #/, ..., xp as zeros, which is contrary (§ 37,
Chap. III.) to the hypothesis that no ^-polynomial vanishes in xli ..., xp.
But a further result follows from the § referred to (§ 158, Chap. VIII.).
Let 2(0iti, ..., %(0i,p, 2&)/( n ..., 2&>/)p denote the periods of ux'a, and
m1} . .., mp, m^', ..., mp' denote any rational integers which are the same for
all values of i. On the hypothesis that the inversion problem is capable of
solution for all values of the quantities U1} ..., Up, suppose these quantities
to vary continuously from the values U1} ..., Up to the values Vlt ..., Vp,
where
Vi = Ui + Zrn^i, l + ...... + ZnipWi, v + 2W/&)/, j + ...... + 2m/o>/> p,
(i = l,2, ...,p),
= Ui+ 2fli, say,
and let zlt ..., zp be the places such that
,,.«,, «i , , uzp<«p = y. .
U>i ilt V I >
then it follows from § 158, that the places zlt . .., zp are, in some order, the
same as the places xl} ...,xp. For this reason it is proper to write the
equations of the inversion problem in the form
where the sign == indicates that the two sides of the congruence differ by a
quantity of the form 2Hj. And further, if the set xlt ...,xp be uniquely
determined by the values Ult ..., Up, any symmetrical function of the values
of x, y at the places of this set, must be a single-valued function of
Ui, ..., Up. Denoting such a function by <j)(Ul, ..., Up), we have, therefore,
</>(tr1 + 2n1, u2+m,, ..., up + 2ap) = <t>(Ul, ..., Up).
The functions that arise are therefore such as are unaltered when the
p variables Ui, ..., Up are simultaneously increased by the same integral
multiples of any one of the 2p sets of quantities denoted by
?Wj pi *^t n •••} ^^p, r
2ft)/, r, 2Q)/, r, . . . , 2ft)/, ,.. (r = 1, 2, . . . , p).
165. The sign = will often be employed in what follows, in the sense
explained above. There is one case in which it is absolutely necessary.
In what has preceded the paths of integration have not been restricted from
crossing the period loops. But it is often convenient, for the sake of
definiteness, to use only integrals for which this restriction is enforced. In
such case the problem expressed by the equations
166] OF JACOBl'S INVERSION PROBLEM. 237
may be incapable of solution for some values of Ul} ..., Up. This can be
seen as follows : if both the sets of equations
were capable of solution, it would follow, by § 158, that the set zlt ..., zp is
the same as the set xlt ... , xp. And thence, as the paths are restricted not
to cross the period loops, we should have
i i i i '
and thence
i p + 2m/&)/) , + ...... + 2wp'a>i'> p = 0 ;
but these equations are reducible to
mi + w/r^ j + ...... + m/Tj, p = 0,
and, therefore, there would exist a function, expressed by
(where v*' a, ..., v^ a are Riemann's elementary integrals of the first kind),
everywhere finite and without periods. Such a function must be a constant ;
thus the conclusion would involve that v*' a, ..., vx' a are not linearly inde
pendent, which is untrue.
Hence when the paths of integration are restricted not to cross the period
loops, the equations of the inversion problem must be written
in this case the integral sum on the left-hand side is not capable of assuming
all values; and the particular period which must be added to the right-hand
side to make the two sides of the congruence equal is determined by the
solution of the problem.
166. Before passing to the proof that Jacobi's inversion problem does
admit of solution, another point should be referred to. It is not at first
sight apparent why it is necessary to take p arguments, Ult ..., Up, and
p dependent places xl} ..., xp. It may be thought, perhaps, that a single
equation
ux- a = U,
wherein ux>* is any definite integral of the first kind, suffices to determine the
place a; as a function of the argument U. We defer to a subsequent place
the enquiry whether this is true when the path of integration on the left
hand is not allowed to cross the period loops of the Riemann surface ; it is
obvious enough that in such a case all conceivable values of U would not arise,
238 JACOBl'S INVERSION PROBLEM. [166
for instance U = oo would not arise, and the function of U obtained would
only be defined for restricted values of the argument. But it is possible
to see that when the path of integration is not limited, the place x cannot be
definitely determinate from U. For, then, putting x =/( U}, we must have
f(U+ 2ft) =f(U), wherein
TOI, ...,mp' being arbitrary rational integers, and 2o>i, . .., 2cop' being the
periods of ux> a ; and it can be shewn, when p > 1, that in general it is
possible to choose the integers TOJ, ..., mp' so that H shall be within assigned
nearness of any prescribed arbitrary value whatever. Thus not only would
the function y(^0 have infinitesimal periods, but any assigned value of this
function would arise for values of the argument lying within assigned near
ness of any value whatever. We shall deal later with the possibility of the
existence of infinitesimal periods; for the present such functions are excluded
from consideration.
The arithmetical theorem referred to* may be described thus; if al} a2
be any real quantities, the values assumed by the expression ^(11+ JV2a2)
when Nlt N2 take all possible rational integer values independently of one
another, are in general infinite in number ; exception arises only in the case
when the ratio a^a2 is rational ; and it is in general possible to find rational
integer values of NI and Nz to make Nlal+N2a2 approach within assigned
nearness of any prescribed real quantity. Similarly if a1} a2, a3, b1} b2, b3 be
real quantities, of the expressions N1a1 + NzOi + Nsa3) N^+N^ + N^s,
where NI, N2, N3 take all possible rational integer values independently
of one another, there are, in general, values which lie within assigned
nearness respectively to two arbitrarily assigned real quantities a, b. More
generally, if alt ..., ak, b1} ...,bk, ...... , clt ...,ck be any (k-1) sets each of
k real quantities, and a, b, ...,c be (k — 1) arbitrary real quantities, it is
in general possible to find rational integers N1} ..., Nk such that the (k — 1)
quantities
N,a, + ...... + Nkak-a, N& + ...... + Nkbk-b, ..., N.C. + ...... +Nkck-c,
are all within assigned nearness of zero.
Hence it follows, taking k = 2j9, that we can choose values of the integers
TOJ , . . . , mp', to make p — 1 of the quantities
nr = TO!&)r)1 + ...... + mpwrip + m1'cor'tl+ ...... +mp'a>r',p,
say Ilj, ..., £lp-i, approach within assigned nearness of any (p— 1) prescribed
values, and at the same time to make the real part of the remaining quantity
£lp approach within assigned nearness of any prescribed value ; but the
imaginary part of £lp will thereby be determined. We cannot therefore
* Jacobi, loc. cit. ; Hermite, CrclJe, LXXXVIII. p. 10.
168] EXISTENCE OF A SOLUTION. 239
expect to obtain an intelligible inversion by taking less than p new variables
Ui, U2, ... ; and it is manifest that we ought to use the same number of
dependent places asl , x2, ____ On the other hand, the proof which has been
given that there can in general only be one set of places aslf ...,xp corre
sponding to given values of Ul, ..., Up would not remain valid in case the
left-hand sides of the equations of the problem of inversion consisted of a
sum of more than p integrals; for it is generally possible to construct a
rational function with p + 1 assigned poles.
167. It follows from the argument here that when p > 1 an integral of the first kind,
ux>a, is capable, for given positions of the extreme limits, x, a, of the integration, of
assuming values within assigned nearness of any prescribed value whatever. Though not
directly connected with the subject here dealt with it is worth remark that it does not
thence follow that the integral is capable of assuming all possible values. For the values
represented by an expression of the form
for all values of the integers m1, ..., mp, m^, ..., mp, form an enumerable aggregate —
that is, they can be arranged in order and numbered — o> , . . . , - 3, -2, — 1 , 0, 1 , 2, 3, . . . , oo .
To prove this we may begin by proving that all values of the form m^ + ragtog form
an enumerable aggregate ; the proof is identical with the proof that all rational fractions
form an enumerable aggregate ; and may then proceed to shew that all values of the form
Mj^+m^ + TOgCBj form an enumerable aggregate, and so on, step by step. Since then the
aggregate of all conceivable complex values is not an enumerable aggregate, the statement
made is justified.
The reader may consult Harkness and Morley, Theory of Functions, p. 280, Dini,
Theorie der Functionen einer reellen Orosse (German edition by Luroth and Schepp),
pp. 27, 191, Cantor, Ada Math. II. pp. 363 — 371, Cantor, Crelle, LXXVII. p. 258, Rendiconti
del Circolo Mat. di Palermo, 1888, pp. 197, 135, 150, where also will be found a theorem
of Poincare's to the effect that no multiform analytical function exists whose values are not
enumerable.
168. Consider now* the equations
(A) <"*' + ...... + u?"a»=Ui, (i=l,2,...,p)
wherein, denoting the differential coefficient of uf a in regard to the infini
tesimal at # by fa (x), the fixed places c^, ..., Op are supposed to be such that
the determinant of p rows and columns whose (i, j)ih element is //,_,- (af) does
not vanish ; wherein also the p paths of integration a1toa;li...,apioxpt are
to be the same in all the p equations, and are not restricted from crossing the
period loops.
When a?!, ..., xp are respectively in the neighbourhoods of a,, ..., ap and
l/i, ..., Up are small, these equations can be written
The argument of this section is derived from Weierstrass ; see the references given in
connection with § 170.
240 CONSTRUCTION OF SOLUTION [168
wherein tf is the infinitesimal in the neighbourhood of the place ar, and //./(#)
is derived from ^r (x) by differentiation. From these equations we obtain
*r=iV,iZ7i + ...... + Vr,PUp+U?+ Uf}+ ...... , (r = l,2, ...,j>),
where, if A denote the determinant whose (i, j)ih element is ^(di), V{j
denotes the minor of this element divided by A, and U^ denotes a homo
geneous integral polynomial in Ult ..., Up of the kih degree. These series
will converge provided U1} ..., Up be of sufficient, not unlimited, smallness.
Hence also, so long as the place xr lies within a certain finite neighbourhood
of the place cr> the values of the variables xr, yr associated with this place,
which are expressible by convergent series of integral powers of tr, are
expressible by series of integral powers of t/j , . . . , Up which are convergent
for sufficiently small values of Uly ..., Up.
Suppose that the values of Ult ...t Up are such that the places xl} ..., xp
thus obtained are not such that the determinant whose (i, j)ih element is
/AJ (xi) is zero ; then if [/"/, . . . , Up be small quantities, it is similarly possible
to obtain p places #/, . . . , xp, lying respectively in the neighbourhoods of
X-L, ..., xp, such that
fcf'-f ...... + <"''*"=£//, (i-l,2,...,p);
by adding these equations to the former we therefore obtain
<•'•"' 4- ...... +U?''0* = Ui + Ut', (i = I, 2, ..., p).
Since all the series used have a finite range of convergence, we are thus
able, step by step, to obtain places xl , . . . , xv to satisfy the p equations
for any finite values of the quantities Ul} ..., Up which can be reached from
the values 0, 0, . . . , 0 without passing through any set of values for which
the corresponding positions of xlt ..., xp render a certain determinant zero.
169. The method of continuation thus sketched has a certain interest;
but we can arrive at the required conclusion in a different way. Let
Ult ..., Uploe any finite quantities ; and let m be a positive integer. When
m is large enough, the quantities UJm, . . . , Upjm are, in absolute value, as
small as we please. Hence there exist places zlt ..., zp, lying respectively in
the neighbourhoods of the places alt ..., ap, such that
M*"a'+ ...... +uz*'ap = - Ui/m (i=I, 2, ...,p).
In order then to obtain places x1} ..., ocp, to satisfy the equations
169] FOR ALL VALUES OF THE ARGUMENTS. 241
it is only necessary to obtain places acl, ...,xp> such that
and it has been shewn (Chap. VIII. § 158), that these equations express only
that the set ofmp + p places formed of zl} ..., zp, each m times repeated and
the places xl} ..., xp, are coresidual with the set of (m + l)p places formed of
Oj, ..., ap each (ra+ 1) times repeated.
Now, when (m + 1)^> places are not zeros of a ^-polynomial, we may
(Chap. VI.) arbitrarily assign all but p of the places of a set of (m+ l)p
places which are coresidual with them; and the other p places will be
algebraically and rationally determinable from the mp assigned places.
Hence with the general positions assigned to the places al} ..., ap, it
follows, if Z denote any rational function, that the values of Z at the places
#!, ..., xp are the roots of an algebraical equation,
whose coefficients El,...,Ep are rationally determinable from the places
z\, •••, Zp, and are therefore, by what has been shewn, expressible by series
of integral powers of UJm, ..., Upfm, which converge for sufficiently large
values of m. Thus the problem expressed by the equations
is always capable of solution, for any finite values of Ult ..., Up.
It has already been shewn (§ 164), that for general values of Ult ..., Up
the set xl} ...,scp obtained is necessarily unique; the same result follows
from the method of the present article. It is clear in § 164, in what way
exception can arise; to see how a corresponding peculiarity may present
itself in the present article the reader may refer to the concluding result
of § 99 (Chap. VI). (See also Chap. III. § 37, Ex. ii.)
In case the places alt ..., ap in the equations (A) be such that the deter
minant denoted by A vanishes, we may take places blt ...,bp, for which
the corresponding determinant is not zero, and follow the argument of the
text for the equations
'
in which Ff = Ut + ua*' b> + . . . . + ua.p> bp.
1 1
We do not enter into the difficulty arising as to the solution of the in
version problem expressed by the equations (A) in the case where Ult..., Up
have such values that a?lf ..., xp are zeros of a ^-polynomial. This point
is best cleared up by actual examination of the functions which are to
be obtained to express the solution of the problem (cf.* § 171, and
* See also Clebsch and Gordan, Abel. Functnen., pp. 184, 186.
B- 16
242 EXPRESSION OF SOLUTION [169
Props, xiii. and xv., Cor. iii., of Chap. X.). But it should be noticed that
the method of § 168 shews that a solution exists in all cases in which the
fixed places c^, ..., ap do not make the determinant A vanish ; the peculiarity
in the special case is that instead of an unique solution sclt ..., xp, all the
GOT+I sets coresidual with xl} ..., xp are equally solutions, r+1 being the
number of linearly independent ^-polynomials which vanish in xl} ...,xp.
This follows from §§ 154, 158.
170. We consider now how to form functions with which to express the
solution of the inversion problem.
Let I* * denote any elementary integral of the third kind, with infinities
at the arbitrary fixed places £ 7. Then if a1} ..., av, xlt ..., asp denote the
places occurring on the left hand in equation (A), it can be shewn that the
function
m_ pxl,a1, pxp,ap
*7t,V ' ' + -tl,y
is the logarithm of a single valued function of Ult ..., Up, and that the
solution of the inversion problem can be expressed by this function ; and
further that, if /*> a denote any Abelian integral, the sum
2*1 1 «i i i jXp, Op
can also* be expressed by the function T.
It is clear that in this statement it is immaterial what integral of the
third kind is adopted. For the difference between two elementary integrals
of the third kind with infinities at £, 7 is of the form
x, a , . -» x, a
1 p p
where \l} ..., \p, \ may depend on f, 7 but are independent of ar; hence
the difference between the two corresponding values of T is of the form
and this is a single-valued function of U1} .... Up.
For definiteness we may therefore suppose that Px' a denotes the integral
of the third kind obtained in Chap. IV. (§ 45. Also Chap. VII. § 134).
Then, firstly, when ae1} ...,xp are very near to al5 ..., ap, and Ult ..., Up
are small, T is given by
ij r> «-\ / ^^i
4 U [(«,, f) - (a,, ,)] - +
* The introduction of the function T is, I believe, due to Weierstrass. See Crelle, LII.
p. 285 (1856) and Mathem. Werke (Berh'n, 1894), i. p. 302. The other functions there used are
considered below in Chaps. XL, XIII.
170] BY ABELIAN INTEGRALS. 243
where ti denotes the infinitesimal in the neighbourhood of the place ait c
is an arbitrary place, and the notation is as in § 130, Chap. VII. It is
intended of course that neither of the places £ or 7 is in the neighbourhood
of any of the places aly ...,ap. Now we have shewn that the infinitesimals
Zj, ..., tp are expressible as convergent series in Ult ..., Up. Thus T is also
expressible as a convergent series in Ult ..., Up when Ult ..., Up are
sufficiently small.
Nextly, suppose the places xly . . . , xp are not near to the places al} ...,ap;
determine, as in § 168, places to satisfy the equations
m being a large positive integer; then we shall also have (§ 158, Chap. VIII.)
where Z (x) denotes the rational function which has a pole of the (m + l)th
order at each of the places a1} ..., ap, and has a zero of the mth order at each
of the places z1} ..., zp. The function Z (x) has also a simple zero at each
of the places xl} ...,xp, but this fact is not part of the definition of the
function.
This equation can be written in the form
wherein T0 denotes the sum
a?
It follows by the proof just given that T0 is expressible as a series of
integral powers of the variables UJm, ..., Up/m, which converges for
sufficiently great values of m; and it is easy to see that the expression
^ (£)/<£ (7) is also expressible by series of integral powers of UJm, ..., Up/m.
For let the most general rational function having a pole of the (m + l)th
order in each of a1} ... , ap be of the form
Z(x)=\1Z1(a;)+ ...... +\npZmp(x)+\,
wherein Z^x), ..., Zmp(x) are definite functions, and \, Xlf ..., \mp are
arbitrary constants. Then the expression of the fact that this function
vanishes to the mth order at each of the places zlt ...,zp will consist of
mp equations determining \1} ..., \np rationally and symmetrically in terms
of the places zlt . . . , zv. Hence (by § 108) \, . . . , X^ are expressible as series
of integral powers of UJm, ..., Up(m. Hence Z(g)/Z(y) is expressible
by series of integral powers of UJm, ..., Up/m.
16—2
244 INTRODUCTION TO SOLUTION [170
Hence, for any finite values of U1} ..., Up the function eT is expressible
by series of integral powers of U1} ..., Up. It is also obvious, from the
method of proof adopted, that the series obtained for any set of values of
£/!, ..., Up are independent of the range of values for U1} ..., Up by which
the final values are reached from the initial set 0, 0, . . . , 0 ; so that the
function eT is a single valued function of Ul} ..., Up. The function eT
reduces to unity for the initial set 0, 0, . . . , 0.
171. An actual expression of the function eT, in terms of Ul} ..., Up,
will be obtained in the next chapter (§ 187, Prop. xiii.). We shew here that
if that expression be known, the solution of the inversion problem can
also be given in explicit terms. Let n£* denote the normal elementary
integral of the third kind (Chap. II., § 14). Then if K denote the sum
it follows, as here, that eK is a single valued function of U1} ..., UP1 whose
expression is known when that of eT is known, and conversely. Denote eK by
V(UL, ..., Up; ^,7). Let Z(x) denote any rational function whatever, its
poles being the places y^, ...,<yk; and let the places at which Z(x) takes
an arbitrary value X be denoted by £x, ..., %k. Then, from the equation
(Chap. VIII., § 154),
we obtain *
the left-hand side of this equation has, we have said, a well ascertained
expression, when the values of Ul, ..., Up. the function Z(sc\ and the value
X, are all given ; hence, substituting for X in turn any p independent
values, we can calculate the expression of any symmetrical function of the
quantities
Z(acl), ..., Z(xp),
and this will constitute the complete solution of the inversion problem.
It has been shewn in § 152, Chap. VIII. that any Abelian integral Ix>a
can be written as a sum of elementary integrals of the third kind and of
differential coefficients of such integrals, together with integrals of the first
kind. Hence, when the expression of V(U1} ..., Up\ g, 7) is obtained, that
of the sum
r*n «i i i 72*. <h>
can also be obtained.
* Clebsch u. Gordan, Abeh. Functionen, (1866), p. 175.
172] BY THETA FUNCTIONS. 245
172. The consideration of the function
TT\xi,O'\ i i f~\xP> aP
fc7 f,y '
which is contained in this chapter is to be regarded as of a preliminary
character. It will appear in the next chapter that it is convenient to
consider this function as expressed in terms of another function, the theta
function. It is possible to build up the theta function in an a priori
manner, which is a generalization of that, depending on the equation
whereby, in the elliptic case, the a-function may be supposed derived from
the function g> (it). But this process is laborious, and furnishes only results
which are more easily evident a> posteriori. For this reason we proceed now
immediately to the theta functions ; formulae connecting these functions
with the algebraical integrals so far considered are given in chapters X. XI.
and XIV.
[173
CHAPTER X.
RIEMANN'S THETA FUNCTIONS. GENERAL THEORY.
173. THE theta functions, which are, certainly, the most important
elements of the theory of this volume, were first introduced by Jacobi in
the case of elliptic functions. * They enabled him to express his functions
sn u, en u, dn u, in the form of fractions having the same denominator, the zeros
of this denominator being the common poles of the functions sn u, en u, dn u.
The ratios of the theta functions, expressed as infinite products, were also
used by Abel f. For the case p = 2, similar functions were found by Gopelj,
who was led to his series by generalizing the form in which Hermite had
written the general exponent of Jacobi's series, and by Rosenhain §, who
first forms degenerate theta functions of two variables by multiplying to
gether two theta functions of one variable, led thereto by the remark that
two integrals of the first kind which exist for p = 2, become elliptic integrals
respectively of the first and third kind, when two branch places of the surface
for p = 2, coincide. Both Gopel and Rosenhain have in view the inversion
problem enunciated by Jacobi ; their memoirs contain a large number of
the ideas that have since been applied to more general cases. In the form
in which the theta functions are considered in this chapter they were first
given, for any value of p, by Riemann||. Functions which are quotients
of theta functions had been previously considered by Weierstrass, without
any mention of the theta series, for any hyperelliptic case 11. These functions
occur in the memoir of Rosenhain, for the case p = 2. It will be seen that
* Fundamenta Nova (1829) ; Ges. Werke (Berlin, 1881), Ed. i. See in particular, Dirichlet,
Gediichtnissrede auf Jacobi, loc. cit. Bd. i., p. 14, and Zur Geschichte der Abelschen Trans-
cendenten, loc. cit., Bd. n., p. 516.
t (Euvres (Christiania, 1881), t. i. p. 343 (1827). See also Eisenstein, Crelle, xxxv. (1847),
p. 153, etc. The equation (b) p. 225, of Eisenstein's memoir, is effectively the equation
i Crelle, xxxv. (1847), p. 277.
§ Mem. sav. etrang. xi. (1851), p. 361. The paper is dated 1846.
|| Crelle, LIV. (1857) ; Ges. Werke, p. 81.
1f Crelle, XLVII. (1854); Crelle, LIT. (1856); Ges. Werke, pp. 133, 297.
174] CONVERGENCE OF THE THETA SERIES. 247
the Riemann theta functions are not the most general form possible. The
subsequent development of the general theory is due largely to Weierstrass.
174. In the case p = l, the convergence of the series obtained by Jacobi
depends upon the use of two periods 2<w, 2<o', for the integral of the first
kind, such that the ratio o>'/&> has its imaginary part positive. Then the
quantity q = e w is, in absolute value, less than unity.
Now it is proved by Riemann that if we choose normal integrals of the
first kind v1^' a, ..., v^a,so that v*'a has the periods 0...0, 1, 0, ..., rr>l, ...,Tr>p,
the imaginary part of the quadratic form
<£ = run^+ ...... + Trtrnrz+ ...... + 2rlj2w1w2 + ...... + 2rr>snrns+ ......
is positive* for all real values of the p variables nly ...,np. Hence for all
rational integer values of nl} ..., np, positive or negative, the quantity ein<i)
has its modulus less than unity. Thus, if we write rr> s = pr> s + iKr> s, pr, s
and Kr>s being real, and a1} =61 + iclj ..., ap, = bp + icp, be any p constant
quantities, the modulus of the general term of the p-fold series
r»p= — o
wherein each of the indices Wj, ..., np takes every real integer value
independently of the other indices, is e~L, where
L = - (b^ + ...... + bpnp) + TT (Knn? + ...... + 2/c1>2 r^n, + ...... ),
= -(&1n1+ ...... + bpnp) + -^, say,
where i/r is a real quadratic form in w1} ..., np, which is essentially positive
for all the values of Wj, . .., np considered. When one (or more) of nlt ..., np
is large, L will have the same sign as ty, and will be positive ; and if p, be any
(f\"f
1 + -) ;
P/
/ 7"\ "^
now the series whose general term is f 1 H — ) will be convergent or not
\ ft/
according as the series whose general term is i/r~'x is convergent or not, for
the ratio 1 + - : ^ has the finite limit I///, for large values of n^, ...,np;
P
and the series whose general term is i/r"'* is convergent provided JJL be taken
* The proof is given in Forsyth, Theory of Functions, § 235. If W* a, ... , ?**' a denote a set of
integrals of the first kind such that wx'a has no periods at the b period loops except at br, and
has there the period 1, and <rr, i , . . . , oy, p be the periods of «£• a at the a period loops, the quadratic
function
<rn«i2+
has its imaginary part negative.
248 EXPLANATION OF NOTATION. [174
> ^p. (Jordan, Cours d 'Analyse, Paris, 1893, vol. I., § 318.) Hence the
series whose general terra is
£«,«! + ...... +apnp+iir<}>
is absolutely convergent.
In what follows we shall write 27riur in place of ar and speak of u1} ...,up
as the arguments; we shall denote by un the quantity uln1 4- ...... +upnp,
and by rn2 the quadratic r^n^ + ...... + 2T12w17i2 + ....... Then the Riemann
theta function is defined by the equation
where the sign of summation indicates that each of the indices nlt ...,np
is to take all positive and negative integral values (including zero),
independently of the others. By what has been proved it follows that © (u)
is a single-valued, integral, analytical function of the arguments v^, ..., up.
The notation is borrowed from the theory of matrices (cf. Appendix ii.) ; T is regarded
as representing the symmetrical matrix whose (r, s)th element is rr> g, n as representing
a row, or column, letter, whose elements are n^, ..., np, and u, similarly, as representing
such a letter with u^ , . . . , up as its elements.
It is convenient, with ® (u), to consider a slightly generalized function,
given by
® (u ; q, q'}, or © (u, q) = Ze2iriu (n+9'}+^r (n+q')*+*riq <n+3') .
herein q denotes the set of p quantities ql} ...,qp, and q' denotes the set
of p quantities #/, . . . , qp, and, for instance, u (n + q) denotes the quantity
un + uq', namely
«iWi + ...... + upnp + u^ + ...... + upqp,
and T (n + <?')2 denotes rn2 -f Zrnq + rq"2, namely
(TllWl2 + . . . + 2r1>2 n, w, +...) + 2 % I rr,s nrqs' + (r11?1/2 + . . . + 2r1)2 q.'q,' +...).
=l r=
The quantities qlt ..., qp, q^, ..., qp constitute, in their aggregate, the
characteristic of the function © (u ; q) ; they may have any constant values
whatever ; in the most common case they are each either 0 or ^.
The quantities r^j are the periods of the Riemann normal integrals of the first kind at
the second set of period loops. It is clear however that any symmetrical matrix, o-, which
is such that for real values of £1} .... kp the quadratic form <rk* has its imaginary part
positive, may be equally used instead of T, to form a convergent series of the same form as
the 6 series. And it is worth while to make this remark in order to point out that the
Riemann theta functions are not of as general a character as possible. For such a
symmetrical matrix o- contains \p Q» + l) different quantities, while the periods rr, g are
(Chap. I., § 7), functions of only 3p-3 independent quantities. The difference |^(p + l)
-(3p-3)=£0°-2)(p-3), vanishes for />=2 orp = 3; for p = 4 it is equal to 1, and for
greater values of p is still greater. We shall afterwards be concerned with the more
general theta-functiou here suggested.
175] FUNDAMENTAL IDENTITIES. 249
The function 6 (w) is obviously a generalization of the theta functions used in the
theory of elliptic functions. One of these, for instance, is given by
and the four elliptic theta functions are in fact obtained by putting respectively q, q' = 0, £ ;
=ii; =i,o; =0,0.
175. There are some general properties of the theta functions, imme
diately deducible from the definition given above, which it is desirable to
put down at once for purposes of reference. Unless the contrary is stated it
is always assumed in this chapter that the characteristic consists of half
integers; we may denote it by J&, ..., ^/3P, ^a1} ..., ^ap, or shortly, by
^fi, ^a, where fti, ..., j3p, a1} ..., ap are integers, in the most common case
either 0 or 1. Further we use the abbreviation nTO)TO/, or sometimes only £lm,
to denote the set of p quantities
wherein mlt ..., mp, m/, ..., mp' are 2p constants. When these constants
are integers, the p quantities denoted by Hm are the periods of the p Riemanri
normal integrals of the first kind when the upper limit of the integrals is taken
round a closed curve which is reducible to mt circuits of the period loop 6;
(or mi crossings of the period loop a{) and to m{ circuits of the period
loop at, i being equal to 1, 2, ...,p. (Cf. the diagram Chap. II. p. 21.)
The general element of the set of p quantities denoted by flm, will also
sometimes be denoted by mt + Tim', T» denoting the row of quantities formed
by the iih row of the matrix r. When mlt ...,mp are integers, the quantity
mi + Tim is the period to be associated with the argument Ui .
Then we have the following formulae, (A), (B), (C), (D), (E) :
@ (- u ; ift }a) = e«** @ (u ; ±0, £a), (A).
Thus @(w ; £/3, £a) is an odd or even function of the variables M,, ..., up
according as /3a, ={3^ + ...... + /3pap> is an odd or even integer; in the
former case we say that the characteristic |/S, £a is an odd characteristic, in
the latter case that it is an even characteristic.
The behaviour of the function ® (it) when proper simultaneous periods
are added to the arguments, is given by the formulae immediately following,
wherein r is any one of the numbers 1, 2, ... , p,
© (?/!, . . . , ur + 1, . . . , up ; i& £o) =
® ( 1*1 + Tlf r , U2 + T2, P , . . . , Up + Tp> r ] % 0, ± Ct) =
Both these are included in the equation
@ (u + nm; i/S, i«) = e-*ri"»' («+*"»') +«(»««-»n'p) e (M ; i /8, ^ a), (B) ;
250 FUNDAMENTAL IDENTITIES. [175
herein the quantities m^ ..., mp, m/, ..., mp are integers, u + £lm stands for
the p quantities such as ur + mr + m1'rr> 1 + + mpTr>p, and the notation
in the exponent on the right hand is that of the theory of matrices ; thus
for instance m'rm denotes the expression
•£
2, mr' (TV, i m/ + + Tfj p mp'\
r=l
and is the same as the expression denoted by rm'2.
Equation (B) shews that the partial differential coefficients, of the second
order, of the logarithm of @ (u ; £/3, ^a), in regard to ult ..., up, are functions
of u1} ...,up, with 2p sets of simultaneous periods.
Equation (B) is included in another equation ; if each of /3', a! denotes a
row of p integers, we have
® (i* + ifV, ..; 1/3, £ a) = e-i-'CH-tf +*'+*«•) ® (u ; 1/3 + 1/3', }a + &), (C) ;
to obtain equation (B) we have only to put /3/ = 2mr, a/ = 2m/ in equation
(C). If, in the same equation, we put ft' = — /3, a.' = — a, we obtain
® (w - |flp, « ; i/3, £a) = e™ <M-iTa) 6 (M ; 0, 0) = e«ia (M-^T"> 0 (M) ;
from this we infer
© (tt + 1^, .), (D) ;
this is an important equation because it reduces a theta function with any
half-integer characteristic to the theta function of zero characteristic.
Finally, when each of m, m denotes a set of p integers, we have the
equation
® (u ; i/3 + m, |a +m') = e™ ® (M ; i/3, -|a), (E) ;
thus the addition of integers to the quantities |« does not alter the theta
function ©(?*; -|/3, ^a), and the addition of integers to the quantities ^/3
can at most change the sign of the function. Hence all the theta functions
with half-integer characteristics are reducible to the 2^ theta functions which
arise when every element of the characteristic is either 0 or £.
176. We shall verify these equations in order in the most direct way. The method
consists in transforming the exponent of the general term of the series, and arranging the
terms in a new order. This process is legitimate, because, as we have proved, the series is
absolutely convergent.
(A) If in the general term
we change the signs of uly ..., up, the exponent becomes
- n - a + ^a) + ITTT ( - n - a + |a) + nift ( - n — a
176] NUMBER OF ODD AND OF EVEN FUNCTIONS. 251
Since a consists of integers we may write m for -n-a, that is mr= -(nr + ar), for
r= 1, 2, ..., p ; then, since # consists of integers, and therefore e2n^n= 1, the general term
becomes
7n'/3a Z-niu(m+^a)+WT(m+^a)+mft(m+^a),
V ' • O 9
save for the factor em^a, this is of the same form as the general term in the original series,
the summation integers mx, ..., mp replacing nt, ..., np. Thus the result is obvious.
(B) The exponent
27rt (u + m + rm') (n + £ a) + iirr (n + \ of + ni(3 (n + \ a\
wherein m+rm' stands for a row, or column, of p quantities of which the general one is
™r+Tr, i %' + ...... +rr,pwip',
is equal to
2iriu (n + ^a) + inr (n -f ^ a)2 + irifi (n + ^ a) -f Znimn + nima
+ iri (ma - ra'/3) + 2nimn.
Replacing /•nimn by 1 and writing n for n+m', the equation (Bj is obtained.
(C) By the work in (B), replacing m, m' by ^/S7, %a respectively, we obtain
and this is immediately seen to be the same as
This proves the formula (C).
It is obvious that equations (D) are only particular cases of equation (C), and the
equation (E) is immediately obvious.
It follows from the equation (A) that the number of odd theta functions contained in
the formula Q(u; jfft £•) is 2»-1(2P- 1), and therefore that the number of even functions
is22P-2P-1(2P-l), or 2P-!(2P+1).
For the number of odd functions is the same as the number of sets of integers,
•^i) y\-> •'••>xj>->yv-> each either 0 or 1, for which
#13/1 + ...... + x],y)) = an odd integer.
These sets consist, (i), of the solutions of the equation
#12/1 + ...... + ^p - \yv - 1 = an odd integer,
in number, say, f(p- 1), each combined with each of the three sets
(*P, &>) = (<>, 1), (1,0), (0,0),
together with, (ii), the solutions of the equation
^i2/i + ...... +#p_1yp_1 = an even integer,
in number ^p~2-f(p-l), each combined with the set
Thus
^) = 3/(p-l
= 22" ~2 + 2 {22» - * + 2/ (p - 2)} = etc.
Hence the number of even half periods is 2" ~ l (2" + 1 ).
252 THE RIEMANN FUNCTION HAS p ZEROS. [177
177. Suppose now that 61, ..., ep are definite constants, that m denotes a
fixed place of the Riemann surface, and x denotes a variable place of the surface.
, x,m i x,m x,m
We consider p arguments given by ur — vr +er, where vv , . . . , vp are
the Riemann normal integrals of the first kind. Then the function © (it) is
a function of x. By equation (B) it satisfies the conditions
© (u + k) = © (u), © (w, + rrk') = e-27rik' <M+*T*'> © (u),
wherein k denotes a row, or column, of integers k1} ..., kp and k' denotes
a row or column * of integers &/, ..., kp'. As a function of x, the function
© (vx> m + e) cannot, clearly, become infinite, for the arguments vr' + er are
always finite ; but the function does vanish ; we proceed in fact to prove the
fundamental theorem — the function © (vx' m + e) has always p zeros of the
first order or zeros whose aggregate multiplicity is p.
For brevity we denote vr' + er by ur. When the arguments u^, ..., up
are nearly equal to any finite values Ult ..., Up, the function © (u) can
be represented by a series of positive integral powers of the differences
MI— Ult ...,up— Up. Hence the zeros of the function ©(M), = © (if- m + e),
are all of positive integral order. The sum of these orders of zero is there
fore equal to the value of the integral
_— . (d log 0 (M) = ~ . 1 1 <K©/ 0)/@ (u) = ^-. I dx I (du,jdx) (©/(»/©(w)),
wherein the dash denotes a partial differentiation in regard to the argument
u,, and the integral is to be taken round the complete boundary of the p-ply
connected surface on which the function is single- valued, namely round the p
closed curves formed by the sides of the period-pair-loops. (Cf. the diagram,
p. 21.)
Now the values of * , . -^ at two points which are opposite points on
0 (u) dx
a period-loop ar are equal, and in the contour integration the corresponding
values of dx are equal and opposite. Hence the portions of the integral
arising from the two sides of a period-loop ar destroy one another. The
values of •*}" . at two points which are opposite points on a period-loop br
differ by — 2?™, or 0, according as s = r or not.
Hence the part of the integral which arises from the period-loop-pair
(ar, br) is equal to — I dur, taken once positively round the left-hand side of
the loop br, namely equal to — (— 1) = 1.
The whole value of the integral is, therefore, p ; this is then the sum
of the orders of zero of the function © (vx> m + e),
* The notation ur + rrk' denotes the p arguments UJ + TJ&', ..., up + rpk'.
178]
EQUATIONS FOR THE POSITION OF THE ZEROS.
253
178. In regard to the position of the zeros of this function we are able
to make some statement. We consider first the case when there are p dis
tinct zeros, each of the first order. It is convenient to dissect the Riemann
surface in such a way that the function log © (vx> m + e) may be regarded as
single-valued on the dissected surface. Denoting the p zeros of © (if- m + e)
by zl} ..., zp, we may suppose the dissection made by p closed curves such as
the one represented in Figure [2], so that a zero of © (if- m + e) is associated
with every one of the period-loop-pairs. Then the surface is still ^>-ply
connected, and log © (u) is single-valued on the surface bounded by the
Fig. 2.
p closed curves such as the one in the figure. For we proved that a com
plete circuit of the closed curve formed by the sides of the (ar, br) period-
loop-pair, gives an increment of 2m for the function log © (u) ; when the
surface is dissected as in the figure this increment of 2jri is again destroyed
in the circuit of the loop which encloses the point zr. Any closed circuit
on the surface as now dissected is equivalent to an aggregate of repetitions of
such circuits as that in the figure ; thus if x be taken round any closed
circuit the value of log © (u) at the conclusion of that circuit will be the
same as at the beginning. From the formulae
which we express by the statement that © (u) has the factors unity and
e-^(ur+^r) for the period loops ar and br respectively, it follows that log@(w)
can, at most, have, for opposite points of ar, br, respectively, differences of
the form ^irigr, - 2m (ur + %rr,r} - 2irihr, wherein gr and hr are integers.
The sides of the loops for which these increments occur are marked in the
figure, ur denoting the value of v*' m + er at the side opposite to that where
254 INVESTIGATION OF THE POSITION [178
the increment is marked; thus ur + ^Tr>r is the mean of the values, ur and
ur + Tr,r> which the integral ur takes at the two sides of the loop br.
Since log ® (u) is now single- valued, the integral - — . / log ® (u) . dus,
£TTI J
taken round all the p closed curves constituting the boundary of the surface,
will have the value zero. Consider the value of this integral taken round the
single boundary in the figure. Let Ar denote the point where the loops
ar, br, and that round zr, meet together. The contribution to the integral
arising from the two sides of ar will be I ffrdv,' m, this integral being taken
once positively round the left side of ar, from Ar back to Ar. This contri
bution is equal to grrr> „. The contribution to the integral =— . I log © (u) dus
ATTlJ
which arises from the two sides of the loop br is equal to
- I
dv]
, m
taken once positively round the left side of the curve br, from Ar back to Ar ;
this is equal to
I / x, m i x 7 a;, m / . 7 \ /•
- J (vr + i rr> r) dvs + (er + hr)fr> , ,
where fr> s is equal to 1 when r = s, and is otherwise zero. Finally the part
i f
of the integral =- -. I log @ (u) dus, which arises by the circuit of the loop
enclosing the point zr, from Ar back to Ar, in the direction indicated by the
arrow head in the figure, is I 'dv*' m where Ar denotes now a definite point on
J Ar
the boundary of the loop br. If we are careful to retain this signification we
may denote this integral by vzsr' r . When we add the results thus obtained,
for the p boundary curves, taking r in turn equal to 1, 2, ...,p, we obtain
r=l
wherein, on the right hand, the br attached to the integral sign indicates
a circuit once positively round the left side of br from Ar back to Ar ; and if
kg denote the quantity defined by the equation
7 4? I / X>m
Ks= 21 (Vr
r=l J br
which, beside the constants of the surface, depends only on the place m,
we have the result
179] OF THE ZEROS OF RIEMANN'S THETA FUNCTION. 255
179. Suppose now that places m1, ..., mp are chosen to satisfy the
conruences
this is always possible (Chap. IX. §§ 1G8, 169) ; it is not necessary for our
purpose, to prove that only one set* of places ml, ...,mp, satisfies the con
ditions ; these places, beside the fixed constants of the surface, depend only
on the place m. Then, by the equations just obtained, we have
/ Zi,1Tli Zp,Wlp, . _ -. .
e, = -(v, + ...... +vs p); (s=l,2,...,p).
Thus if we express the zero in the function © (if> m + e), it takes the form
©/ x, m Zi. Wi Zn. niv 7 / ,
(vs -vs1' '- ...... -vsp' P-hs'-Tsg'),
where #/, ..., gp', h{, . . . , hp' are certain integers, and this, by the fundamental
equation (B), § 175, is equal to
z'mp
-v/'p),
save for the factor e-W <«*"•-«*•"'- ...... -„*.•*- 4^ This factor does not
vanish or become infinite. Hence we have the result : It is possible, corre
sponding to any place m, to choose p places, ml} ..., mp, whose position depends
only on the position of m, such that the zeros of the function,
(H) (<yS, TO — yz, , nti — ...... _ vzp, mp\
regarded as a function of x, are the places z1} . .., zp. This is a very funda
mental result f.
It is to be noticed that the arguments expressed by vx> m — vZl> mi— ... — vZp> m»
do not in fact depend on the place m. For the equations for m1} ..., mp,
corresponding to any arbitrary position of m, were
mP,Ap _ j £ f , x,
+VS =/Cs, = 2, (Vr
r=lJ br
a being an arbitrary place. If, instead of m, we take another place /*, we
shall, similarly, be required to determine places fjL1} ..., ^p by the equations
^'Ap^ks, = f ! (,r + K,r)^'a,
r=l J br
* If two sets satisfy the conditions, these sets will be coresidual (Chap. VIII., § 158).
t Cf. Riemann, Ges. Werke (1876), p. 125, (§ 22). The places mlf ... , mp are used by Clebsch
u. Gordan (Abel. Functional, 1866), p. 195. In Riemann's arrangement the existence of the
solution of the inversion problem is not proved before the theta functions are introduced.
256 CASE WHEN THE ZEROS [179
thus
Puttii up, mp S. i m, n j x, a
v, + ...... +< =2 vr dvs , =
r=lJ br
r=l
wherein fgj r = 1 when r = s, and is otherwise zero, as we see by recalling
the significance of the br attached to the integral sign. Thus (Chap. VIII.,
§ 158), the places p,l, ..., pp, m are coresidual with the places mlt ..., mp, /*,
and the arguments
x, m z,, m, zp, TOP
« —vs — —fl
congruent to arguments of the form
Zp, ftp
-vt .
The fact that the places fi1, ...,fj,p,m are coresidual with the places
m,i, ..., mp, fji, which is expressed by the equations
Ui , nil Mp, Win Wl. M, ~
•T + ...... +C +fls =0,
will also, in future, be often represented in the form
If the places wil5 ..., mp are not zeros of a ^-polynomial, this relation
determines /AJ , . . . , JJ,P uniquely from the place /A.
Ex. In case j0 = l, prove that the relation determining m1} ..., mp leads to
Hence the function 0 (v*- 2 + ^ + Jr) vanishes for ^p=z, as is otherwise obvious.
180. The deductions so far made, on the supposition that the p zeros of
the function @ (vz> m + e) are distinct, are not essentially modified when this
is not so. Suppose the zeros to consist of a jvtuple zero at zlt a £>2-tuple zero
at zz , ..., and a ^-tuple zero at z^, so that pl -f ...... +pk=P- The surface
may be dissected into a simply connected surface as in Figure 3. The
function log ®(vx>m + e) becomes a single- valued function of # on the
dissected surface ; and its differences, for the two sides of the various cuts,
are those given in the figure. To obtain these differences we remember
that log @ (fl*' m + e) increases by 2m when x is taken completely round
the four sides of a pair of loops (ar, br). The mode of dissection of Fig. 3,
may of course also be used in the previous case when the zeros of ® (vx> m + e)
are all of the first order.
The integral ^— . I log © (vx> ™ + e) dv*' m, taken along the single closed
ZiTTI J
boundary constituted by the sides of all the cuts, has the value zero. Its
180] ARE NOT DISTINCT.
value is, however, in the case of Figure 3,
257
+pkv
k'Al
4- ffl f dvT m - h, ! dvx; * - 1 (tf m + * + KI) dvx; m - (P - i) „
J a\ J b} J 6,
,At
a.
dv
'm-[ (v*'
J 6,
r r r
I j x'm 7 T x, Jtt / , a;,
l ctos -hpl dvs - (vp
J OP J bp J bp
wherein the first row is that obtained by the sides of the cuts, from Al,
excluding the zeros zl} ..., zk, and the second row is that obtained from
the cuts «!, &j, d, and so on. The suffix ^ to the first integral sign in
the second row indicates that the integral is to be taken once positively round
the left side* of the cut #1, the suffix 6j indicates a similar path for the
cut 6j , and so on. If, as before, we put kx for the sum
/ v r / x'm , i \ j x>m
Kg, = 2, (Vr +$Trtr)dvs ,
r=lJ br
we obtain, therefore, as the result of the integration, that the quantity
tl+ + gprs> p + eK
* By the left side of a cut ax , or fcj , is meant the side upon which the increments of log 6 (M)
are marked in the figure. The general question of the effect of variation in the period cuts is
most conveniently postponed until the transformation of the theta functions has been considered.
B. 17
258 THE THETA FUNCTION VANISHES IDENTICALLY [180
is equal to
, Zi, A, zk, AI , , x A?, A , , £.. A3, A., Ap, Ap-i
ks-pivs - ...... -JW + (p-l)v, + (p_2)v8 + ...... + V,
and this is immediately seen to be the same as
We thus obtain, of course, the same equations as before (§ 179), save that
z1 is here repeated pl times, ..., and z% is repeated p^ times. And
we can draw the inference that © (vx> m + e) can be written in the form
© (/•• « _ ^.- «, _ ...... _ /,. »«P _ /is _ Ts^ which, save for a finite non-vanish
ing factor, is the same as @(/>m-^"m'- ...... _/»»w»); the argument
v*' m - /" m' - . . . . - vzp' mp does not depend on the place m.
8 S S •*• A
181. From the results of §§ 179, 180, we can draw an inference which
leads to most important developments in the theory of the theta functions.
For, from what is there obtained it follows that if zl, ..., zp be any places
whatever, the function ®(vx'm - vZi'mi - ...... _^'m») has zlt ..., zp for
zeros. Hence, putting zp for x we infer that the function
vanishes identically for all positions of zl} ..., %_j. Putting
/2i,?«i . Zp-2, Wu-2 m.n.m,
>=VS + ...... +VS
-V
for s= 1, 2, ..., p, this is the same as the statement that the function
(S) (vx> in>>-1 + f) vanishes identically for all positions of a; and for all values
of/j, ...,fp which can be expressed in the form arising here. When/, ...,/,
are arbitrary quantities it is not in general possible to determine places
zl} ...,Zp_2 to express/, ...,/, in the form in question. Nevertheless the
case which presents itself reminds us that in the investigation of the zeros
of <H) (if' m + e) we have assumed that the function does not vanish identically,
and it is essential to observe that this is so for general values of els ..., ep.
If, for a given position of x, the function ® (^ m + e) vanished identically for
all values of ely ..., ep, the function @ (r) would vanish for all values of the
arguments rlt ..., rp. We assume* from the original definition of the theta
function, by means of a series, that this is not the case.
Further the function @ (vx> m + e) is by definition an analytical function of
each of the quantities e1} ..., ep ; and if an analytical function do not vanish
* The series is a series of integral powers of the quantities e2irir', ____ e'mrp_
182] FOR CERTAIN FORMS OF THE ARGUMENT. 259
for all values of its argument, there must exist a continuum of values of
the argument, of finite extent in two dimensions, within which the function
does not vanish*. Hence, for each of the quantities ely ...,ep there is a
continuum of values of two dimensions, within which the function @ («*> m + e)
does not vanish identically. And, by equation (B), § 175, this statement
remains true when the quantities e1} ...,ep are increased by any simultaneous
periods. Restricting ourselves then, first of all, to values of e1} ..., ep lying
within these regions, there exist (Chap. IX. § 168) positions of zl} ..., zp to
satisfy the congruences
and, since to each set of positions of zl} ..., zp, there corresponds only one set
of values for elf ..., ep> the places zlt ..., zp are also, each of them, variable
within a certain two-dimensionality. Hence, within certain two-dimensional
limits, there certainly exist arbitrary values of zlt ..., zp such that the function
®(v*'m -vz"m> - ...... _w*>«*) does not vanish identically. For such
values, and the corresponding values of elt ..., ep, the investigation so
far given holds good. And therefore, for such values, the function
® (vm»< '«_/..'«._ ...... _ ^P-I, mp-i) vanishes identically. Since this function
is an analytical function of the placesf ^, ..., zp_lt and vanishes identically
for all positions of each of these places within a certain continuum of two
dimensions, it must vanish identically for all positions of these places.
Hence the theorem (F) holds without limitation, notwithstanding the
fact that for certain special forms of the quantities elt ...,ep, the function
<H) (^ m + e) vanishes identically. The important part played by the theorem
(F) will be seen to justify this enquiry.
J82. It is convenient now to deduce in order a series of propositions in
regard to the theta functions (§§ 182—188); and for purposes of reference
it is desirable to number them.
(I.) If £1; ...,%p be p places which are zeros of one or more linearly
independent ^-polynomials, that is, of linearly independent linear aggregates
of the form XIHI(«)+ ...... + \p£lp(x) (Chap. II. § 18, Chap. VI. § 101), then
the function
vanishes identically for all positions of x.
For then, if r + 1 be the number of linearly independent ^-polynomials
which vanish in the places £,,...,£,, we can, taking r + 1 arbitrary places
* E.g. a single-valued analytical function of an argument 2, =x + iy, cannot vanish for all
rational values of x and y without vanishing identically.
t By an analytical function of a place 2 on a Riemann surface, is meant a function whose
values can be expressed by series of integral powers of the infinitesimal at the place.
17—2
260 SUMMARY OF RESULT. [182
zl} ..., zT+l, determine p— r— 1 places zT+2, ..., zp, such that (zlt ..., zp)
= (&, ...,£P) (see Chap. VI. § 93, etc., and for the notation, § 179). Then the
argument
t?> m _ 7,f" m> _ _Jp>mP (<i — ~( 2 V\
Us — Vg — Vg > V6 ~~ L> ^> • ' • ' P/>
can be put in the form
save for integral multiples of the periods ; thus (§§ 179, 180) the theta
function vanishes when a; is at any one of the perfectly arbitrary places
zlt ..., ZT+I. Thus, since by hypothesis r+ 1 is at least equal to 1, the theta
function vanishes identically.
It follows from this proposition that if z2', ..., zp' be the remaining zeros
of a (^-polynomial determined to vanish in each of z2, ...,zp, and neither
x nor z-i be among z2'> • • • > Zp, then the zeros of the function
regarded as a function of zlt are the places x, zj , ..., zp.
From this Proposition and the results previously obtained, we can infer
that the function ®(v*'m — vZl'm> — ...... —vz>>'m") vanishes only (i) when a;
coincides with one of the places zl} ..., zp, or (ii) when zlt ..., zp are zeros of
a ^-polynomial.
(II.) Suppose a rational function exists, of order, Q, not greater than p,
and let T + 1 be the number of (^-polynomials vanishing in the poles of this
function. Take r + 1 arbitrary places
Sl) •••> b<?> «^i> •••» ^T+l—qy
wherein q= Q — p +r+l, and suppose zlt ..., z(t to be a set of places core-
sidual with the poles of the rational function, of which, therefore, q are
arbitrary. Then the function
_ _ yZq+l , »«T+2-q _ ^ _ _ _ ^Z«> mt>-Q\
vanishes identically.
For if we choose £q+l, ..., %Q such that (%lt ..., £e) = (zlt ..., ZQ), the
general argument of the theta function under consideration is congruent
to the argument
nip, m xl,m1 av-t-i-g. mT+i-q q+i, mr+z-q Q, mp-q
This value of the argument is a particular case of that occurring in
(F), § 181, the last q— 1 of the upper limits in (F) being put equal to the
lower limits. Hence the proposition follows from (F).
182] A PARTICULAR FORM OF THE INVERSION PROBLEM. 261
(III.) If r denote such a set of arguments r,, ..., rp that 0 (r) = 0, and,
for the positions of z under consideration, the function ® (vx> z + r) does not
vanish for all positions of as, then there are unique places zlt ..., ^_1(
such that
r - ymp, m _ vzt , mt _ ^ ^ _ rfp-i, mp-i
In this statement of the proposition a further abbreviation is introduced
which will be constantly employed. The suffix indicating that the equation
stands as the representative of p equations is omitted.
Before proceeding to the proof it may be remarked that if m', m/, ...,mp'
be places such that (cf. § 179)
(m', m1} ..., mp) = (m, w/, ..., mp'}
and therefore, also,
then the equation
ym', 7/1 _ vm,', »«i _ ^ ^ _ vm,,', m,,
r = vm>" m
is the same as the equation
r = v
This proposition (III.) is in the nature of a converse to equation (F).
Since the function @ (vx' z -f r) does not vanish identically, its zeros, z1} ..., zp,
are such that
vx'z+r=vx'm-vZt'Wl- ...... _/"•'«";
now we have
vz» m' + vz»> mp = vz»- '"' + vz» m"
so that the zeros zlt ..., zp may be taken in any order ; since ® (r) vanishes,
z is one of the zeros of % (vx> z + r); hence, we may put zp = z, and obtain
r = . _ '• .
_ m,,, m zlt m,
which is the form in question.
If the places zlt ...,zp_l in this equation are not unique, but, on the
contrary, there exists also an equation of the form
r = v™1" m — v*1
then, from the resulting equation
262 A PARTICULAR FORM [182
we can (Chap. VIII. § 158) infer that there is an infinite number of sets of
places Zi, ..., z'p-!, all coresidual with the set zl} ..., zp_-^ ; hence we can put
wherein at least one of the places z^, . . . , zfp^l is entirely arbitrary. Then the
function © (vx> z + r) vanishes for an arbitrary position of x, that is, it
vanishes identically ; this is contrary to the hypothesis made.
It follows also that whenever it is possible to find places ^ , . . . , zp^l to
satisfy the inversion problem expressed by the p equations
the function © (vm>" m — u) vanishes ; conversely, when u is such that this
function vanishes we can solve the inversion problem referred to.
(IV.) When r is such that © (r) vanishes, and © (vx> z + r) does not,
for the values of z considered, vanish identically for all positions of a, the
zeros of © (vx> z + r), other than z, are independent of z and depend only on
the argument r.
This is an immediate corollary from Proposition (III.) ; but it is of
sufficient importance to be stated separately.
(V.) If © (r) = 0, and © (vx> z + r) vanish identically for all positions
of x and z, but © (vx< z 4- tf' ^ + r) do not vanish identically, in regard to x,
for the positions of z, £, % considered, then it is possible to find places
z1} ..., Zp-2 such that
m"' m — Zl' m'
and these places z1) ..., Zp_,, are definite.
Under the hypotheses made, we can put
r =
wherein zlt . . . , zp are the zeros of © (if- z + v*' * + r) ; now z is clearly a zero ;
for the function © (v*'*+ r) is of the same form as © (v*>z + r), and vanishes
identically; and fis also a zero; for, putting ffor#, the function ®(vx'z+vt>t+r)
becomes © (v* > z + r), which also vanishes identically. Putting, therefore, f, z
for ,Zp_i and zp respectively, the result enunciated is obtained, the uniqueness
of the places zl} ..., Zp_2 being inferred as in Proposition (III.).
We may state the theorem differently thus : If © (vx> z + r) vanish for
all positions of x and z, and @ (vx> z + v*> * + r) do not in general vanish
identically, the equations
r = v
182] OF JACOBI'S INVERSION PROBLEM. 263
can be solved, and in the solution one of z1} . .., Zp__^ may be taken arbitrarily,
and the others are thereby determined. Hence also we can find places
Zi, ..., z'p-i, other than z1} ..., zp_l, such that
one of the places £/, ..., z'p_l being arbitrary. Hence by the formula
Q-q = p — T — 1, putting Q=p—lt q=l, we infer r+l=2, so that a
0-polynomial vanishing in zlt ..., zp^ can be made to vanish in the further
arbitrary place z. Thus, when © («*> z + r) vanishes identically, we can write
/• * + r = v*> "< _ /.. «». _ ...... _ ^'->. «P-« _ /• »*
wherein the places ^, ..., ^_1} z are zeros of a ^-polynomial (cf. Prop. I.).
(VI.) The propositions (III.) and (V.) can be generalized thus : If
© (vx>'Zt + ...... + vx"'Zq + r) be identically zero for all positions of the places
xl,zl, ...,xq, zq, and the function © (vx'z + vx"z' + ...... +vx<i'z'' + r) do not
vanish identically in regard to x, then places £i, ..., ^p_l can be found to
satisfy the equations
r = v
and, of these places, q are arbitrary, the others being thereby determined.
These arbitrary places, £,, ..., %q, say, must be such that the function
© (/' s + /" Zl + ...... + v*"' Zq + r) does not vanish identically.
For as before we can put
, z
wherein ^ , . . . , fp are the zeros of the function © (vx> z + v*1' Zl + . . . + v*" Zq + r).
It is clear that z is one zero of this function ; also putting z1 for x the function
becomes @ (vx"z + tf" 2s + ...... + wx" Zq + r), which vanishes, by the hypothesis.
Thus the places z, z1} ..., zq are all zeros of the function
Putting then z,, ..., gq, z respectively for f,, ..., £q, £p in the congruence
just written, it becomes
and this is the same as
replacing #1, .... a;7 by £,,..., f, we have the result stated.
264 PROPERTIES OF THE PLACES [182
Hence also, we can tiud places £/, . .., £p-l} other than £1; ..., ^p_l, such
that
q of the places £/, ..., £'p-i being arbitrary. Therefore a (^-polynomial can
be chosen to vanish in ^ , . . . , ^p_1 and in q (= p — 1 — (Q — q), when Q=p— 1)
other arbitrary places. Thus the argument
for which the theta function vanishes identically, can be written in the form
> m»
wherein .Zj, ...,zq_l} ^IJt . .., %P-I> z are zeros of q+l linearly independent
(^-polynomials.
(VII.) If the function © (/" Zl + ...... + vx"' Zl' + r) be identically zero for
all positions of the places x1, zly x2, z2, ..., xq, zq, and, for general positions of
aslt z,, ..., seq, zq, the function @ (/' z + v*1' z> + ...... +vx"'Zq+r) be not
identically zero, as a function of x, for proper positions of z, and be not
identically zero, as a function of z, for proper positions of x, then we can find
places £i, ..., £p-lt of which q places are arbitrary, such that
and can also find places £1} ..., £p_i, of which q places are arbitrary, such
that
— r = vmp> m — v*1 ' m' — . . — v*1" lf m"'\
This is obvious from the last proposition, if we notice that
We can hence infer that
2vmp' m + vmi'^ + vmi'*l + ...... -f v"1"-1' &~1 + vmp~1' fp~1 = 0,
and this is the same (Chap. VIII. § 158) as the statement that the set of
2p places constituted by &,..., %p-i, &, ..., ^,_i and the place m, repeated, is
coresidual with the set of 2p places constituted by the places m^, ..., mp, each
repeated. This result we write (cf. § 179) in the form
(m2, £, ... , ^_1} f,, ..., ^j) = (m^, mj, ..., m/}.
(VIII.) We can now prove that if £, ..., ^_j be arbitrary places, places
£1} ..., £p_j can be found such that
(m2, f !,..., fp_i, ?!,..., fp-O = (wh3, w22, ..., wip3).
Let r denote the set of j9 arguments given by
182] mlt nit, ..., mp. 265
£i> •••> £p-i being quite arbitrary. Then, by theorem (F), (§ 181), the function
® (r) certainly vanishes. It may happen that also the function ® (vx- z + r)
vanishes identically for all positions of # and z. It may further happen that
also the function S («*• z + if1' Zi + r) vanishes identically for all positions of
x, z, #!, z1. We assume* however that there is a finite value of q such that
the function © (vx' z + vx>'Zl + ...... + /'" * + r) does not vanish identically for
all positions of x, z, aslt zl} . .., xq, zq. Then by Proposition VII. it follows
that we can find places £lf ..., gp_lt such that
— r = v
= ™1"
comparing this with the equations defining the argument r, we can, as
in Proposition (VII.) infer that the congruence stated at the beginning of
this Proposition also holds.
(IX.) Hence follows a very important corollary. Taking any other
arbitrary places £/, ..., ^'p-lt we can find places £/, ..., ^'^ such that
(m2, £', ..., f'^, £/, ..., £'^) = (m*, m22, ..., m/);
therefore the set £, ..., ^p_lt £, ....f^ is coresidual with the set £', ...,^_1,
£/> •••» ^"p-i- Now, of a set of 2p — 2 places coresidual with a given set
we can in general take only p — 2 arbitrarily ; when, as here, we can take
p — 1 arbitrarily, each of the sets must be the zeros of a ^-polynomial
(Chap. VI. § 93). Thus the places £, ..., £,_,, £, ..., ^ are zeros of a
</>-polynomial.
Therefore, if a1} ..., O2p_2 be the zeros of any ^-polynomial whatever,
that is, the zeros of the differential of any integral of the first kind, the
places m^, ... , mp are so derived from the place m that we have
(w2, a1} . . . , a2i,_2) = (m,-, w22, . . . , m/), (G) ;
in other words, if c1} . . . , cp denote any independent places, the places ma, mp
satisfy the equations
2 /'' ' Cl + ...... + v™>" e" = 2l) c" "1' c>1 ' Cl *2p-3' Cp "2"-2' c"
for s = 1, 2, . . . , p. Denoting the right hand, whose value is perfectly definite,
by Ag, and supposing ffl, ..., ffpt h,, ..., hp to denote proper integers, these
equations are the same as
C'C' + ...... + C>Cj>=^. + H*. + flriT.|1 + ...... +ffPT.,p), (G'),
where 8=1, 2, ...,p.
* It will be seen in Proposition XIV. that if 0 (v*> z+vx» z' + ...... +vx*<z' + r) vanishes
identically, then all the partial differential coefficients of 9 (M), in regard to i^, ... , up, up to and
including those of the (q + l)th order, also vanish for u = r.
266 GEOMETRICAL INTERPRETATION [182
There are however 2'-* sets of places -m^, ...,mp, corresponding to any
position of the place m, which satisfy the equation* (G). For in equations
(G') there are 22^ values possible for the right-hand side in which each
of (ft, . .. , gp, hlt . .., hp is either 0 or 1, and any two sets of values glt ..., gp,
h1} ..., hp and #/, ..., gp', A/, ..., hp', such that g^g{ differ by an even integer,
and hi, A/ differ by an even integer, for i = l, 2, ...,p, lead to the same
positions for the places mlt ..., mp. (Chap. VIII. § 158.)
We have seen (§ 179) that the places m1} ...,mp depend only on the place
m and on the mode of dissection of the Riemann surface. We are to see,
in what follows, that the 22p solutions of the equation (G) are to be associated,
in an unique way, each with one of the 2-p essentially distinct theta functions
with half integer characteristics.
183. The equation (G) can be interpreted geometrically. Take a rion-
adjoint polynomial, A, of any grade /JL, which has a zero of the second order
at the place m ; it will have tip — 2 other zeros. Take an adjoint polynomial
>|r, of grade (n — 1) 0 + n— 3 + p, which vanishes in these other n/j,— 2 zeros
of A. Then (Chap. VI. § 92, Ex. ix.) ^ will be of the form X^0 + A<£,
where i/r0 is a special form of ty, A, is an arbitrary constant, and <f> is a
general (^-polynomial. The polynomial i/r will have Zp zeros other than
those prescribed ; denote them by klt . . . , k.^. If </>' be any ^-polynomial, with
a1} ..., a2p-2 as zeros, we can form a rational function, given by (Xi/r0+A</>)/A<£',
whose poles are the places al, ..., a^-z, together with the place m repeated,
its zeros being the places kl} ..., kzp. Hence (Chap. VI. § 96) we have
\ni~, a>i, ..., a.2p—2)^(K1, ** '••> "'zp— 1> kzp),
and therefore, by equation (G),
(nij1, ... , mp-} = (&j , ki, ..., £sp_i , k2p) (G") ;
hence (Chap. VI. § 90) it is possible to take the polynomial -fy so that
its zeros kly ..., k2p consist of p zeros each of the second order, and the
places nil, ..., mp are one of the sets of p places thus obtained.
There are 22p possible polynomials ty which have the necessary character,
as we have already seen by considering the equation (G'); but, in fact,
a certain number of these are composite polynomials formed by the product
of the polynomial A and a ^-polynomial of which the 2p — 2 zeros consist of
p - 1 zeros each repeated. To prove this it is sufficient to prove that there
exist such ^-polynomials having only p — 1 zeros, each of the second order ;
for it is clear that if <£ denote such a polynomial, the product A<I> is of grade
* If for any set of values for gl , . . . , gp , /ij , . . . , hp the equations (G') are capable of an infinity
of (coresidual) sets of solutions, the correct statement will be that there are 22P lots of coresidual
sets, belonging to the place m, which satisfy the equation (G). The corresponding modification
may be made in what follows.
184] OF THESE PLACES. 267
(ft — l)a + n — 3 + /u, and satisfies the conditions imposed on the polynomial -ty.
That there are such (^-polynomials <I> is immediately obvious algebraically.
If we form the equation giving the values of x at the zeros of the general
(^-polynomial,
the p — 1 conditions that the left-hand side should be a perfect square, will
determine the necessary ratios Xj : X2 : ... : \p, and, in general, in only a
finite number of ways. (Cf. also Prop. XL below.)
It is immediately seen, from equation (G"), that if m^ , ..., mp be the
double zeros of one such polynomial -fy- as described, and w/, . . . , mp' of
another, both sets being derived from the same place ra, then
vmt''mt + ...... + /'"'' ""' = | flPi . , (H)
where £Lpt tt stands for p quantities such as
«n •••» «?> A. •••! $p being integers.
We may give an example of the geometrical relation thus introduced, which is of great
importance. It will be sufficient to use only the usual geometrical phraseology.
Suppose the fundamental equation is of the form
;/)i + (x, y)jj + (x, y)3 + (#, y\ = 0,
representing a plane quartic curve (p = 3). Then if a straight line be drawn touching the
curve at a point m, it will intersect it again in 2 points A, B. Through these 2 points
A, J3, oo 3 conies can be drawn ; of these conies there are a certain number which touch
the fundamental quartic in three points P, Q, R other than A and B. There are 22* = 64
sets of three such points I1, Q, R ; but of these some consist of the two points of contact
of double tangents of the quartic taken with the point m itself.
In fact there are (Salmon, Higher Plane Curves, Dublin, 1879, p. 213) 28, = 2»-1(2p-l),
double tangents ; these do not depend at all on the point m ; there are therefore
36, =2"-1(2''4-l), proper sets of three points P, Q, R in which conies passing through
A and B touch the curve. One of these sets of three points is formed by the points
wij, m2, m3. It has been proved that the numbers 2" -1 (2"- 1), 2^~J (2^+1) are respectively
the numbers of odd and even theta functions of half integer characteristics (§ 176).
184. (X.) We have seen in Proposition (VIII.) (§ 182) that the places
Wj , . . . , mp are one set from 2s* sets of p places all satisfying the same
equivalence (G). We are now to see the interpretation of the other 22^ — 1
solutions of this equation.
Let w/, ...,mp' be any set, other than ml} ...,mp, which satisfies the
congruence (G). Then, by equations (G'), we have
2 «"''"" + ...... +<"'-'"")^0. (*=1,2, ...,p),
268 THE 2^ POSSIBLE POSITIONS. [184
and therefore, if H^>0 denote the set of p quantities of which a general one is
given by
& + aiT*,i 4- ...... +<*PTgjp> (s = l, 2, ...,p),
where a1} ..., ap, @i, ..., &p are certain integers, we have
Mil', »», mp, ma ! .-.
vs + ...... +v,p P=^n^a;
hence the function
= e®* © (vz< ' mi'
where
the function is therefore equal to
,m,p x, m / -i c\ \
P-w , (s = l, 2, ...,_p);
by equation (C), § 175 ; thus the function ®(vx' m-vz»m> - ...... - vz"> '""' ,
vanishes when x is at either of the places z1, ..., zp.
We can similarly prove that
It has been remarked (§ 175) that there are effectively 22^ theta functions,
corresponding to the 2^ sets of values of the integers a, /3 in which each
is either 0 or 1. The present proposition enables us to associate each of
the functions with one of the solutions of the equivalence (G). When the
function ® (vx> m ; ^(3, ^a) does not vanish identically in respect to x, its
zeros are the places m/, ..., mp' '. Therefore, instead of the function © (u),
we may regard the function ®(u; £/3, ^a) as fundamental, and shall only be
led to the places m^, ..., mp', instead of m1} ...,mp.
(XL) The sets of places m/, . . . , mp which are connected with the places
m1} ..., nip by means of the equations
wherein a1} ..., ap, {3lt ..., /3P denote in turn all the 22p sets of values in which
each element is either 0 or 1, may be divided into two categories, according
as the integer /3a, = fti^ + ...... + @pap , is even or odd. We have remarked,
in Proposition (IX.), that they may be divided into two categories according
as they are the zeros, of the second order, of a proper polynomial X-^0 + A</>,
or consist of the p — 1 zeros, each of the second order, of a ^-polynomial
together with the place m. When the fundamental Riemann surface is
perfectly general these two methods of division of the 22^ sets entirely agree.
When /3a is odd, m^, . . . , mp' consist of the place m and the p - 1 zeros,
each of the second order, of a ^-polynomial. When /3a is even, m/, . . . , mp'
184] ODD AND EVEN CHARACTERISTICS. 269
consist of the zeros, each of the second order, of a proper polynomial >/r. In
the latter case we may speak of the places ?»/, . . . , mp' as a set of tangential
derivatives of the place m.
For by the equations (D), (A), (§ 175), we have
e,iau (B) Qft^ B + M)/g-«-u (H) Qft^ a _ w) = e-nifia .
hence, when /3<z is odd, eniau ® (^flftt a + u) is an odd function of u, and
must vanish when u is zero; since then ® (|n^a) vanishes, there exist, by
Proposition (VII.), places nl} ..., np_l} such that
or
Hence (Chap. VIII. § 158) we have
(m2, w,2, ..., ny,) = (m,2, ..., m/),
so that, by equation (G), the places nlt ..., wp_, are the zeros of a ^-polynomial,
each being of the second order.
When pa. is even, the function e™u ® (£O0, a + u} is an even function, and
it is to be expected that it will not vanish for u = 0. This is generally the
case, but exception may arise when the fundamental Riemann surface is of
special character. We are thus led to make a distinction between the general
case, which, noticing that ® (%&?,* + u) is equal to e~iria(u+^~^Ta} ® (u ; £/3, £a),
may be described as that in which no even theta function vanishes for zero
values of the argument, and special cases in which one or more even theta
functions do vanish for zero values of the argument.
Suppose then, firstly, that no even theta function vanishes for zero values
of the argument. Then if w/, ..., np_^ be places which, repeated, are the
zeros of a ^-polynomial, we have
(m2, w/2, . . . , M/ap_i) = 0»,s, ™22, • • • , ?V) 5
hence the argument
is a half-period, = - £ft/r, a', say. Thus, by the result (F), @ (^^Vy) is zero ;
therefore, by the hypothesis pet is an odd integer. So that, in this case,
every odd half-period corresponds to a </>-polynomial of which all the zeros
are of the second order, and conversely.
Further, in this case it is immediately obvious that the places TW,, ..., mp
do not consist of the place m and the zeros of a ^-polynomial whose zeros are
of the second order ; for if mlt ...,mp were the places n,, ..., np_lt m, then, by
the result (F), the function © (vz"n' + + /"-'• ""-') would vanish for all
positions of z} zp_^ and therefore <") (0) would vanish.
270 EVEN THETA FUNCTIONS MAY VANISH [185
185. If, however, nextly, there be even theta functions which vanish
for zero values of the argument, it does not follow as above that every
^-polynomial with double zeros corresponds to an odd half-period ; there
will still be such ^-polynomials corresponding to the W~l (2^ — 1) odd half-
periods, but there will also be such ^-polynomials corresponding to even
half-periods.
For if fli, ..., ap, fti, ..., ftp be integers such that fta. is even, and
® (w -f- 1 H^ „) vanishes for u = 0, the first differential coefficients, in regard
to ulf ...,up, of the even function e7""" S(u + ^ft^ „), being odd functions,
will vanish for u = 0. By an argument which, for convenience, is postponed
to Prop. XIV., it follows that then the function 0 (v*' z + \ fl^ „) vanishes
identically for all positions of x and z. Therefore, by Prop. V., there is at
least a single infinity of places zl , ..., zp_^ satisfying the equations
- 100, 0^W"'W -/"""- ...... -a*-'.""-' ;
these equations are equivalent to
hence there is a single infinity of ^-polynomials with double zeros corre
sponding to the even half-period ^Ii^) „ , and their p — 1 zeros form coresidual
sets with multiplicity at least equal to 1.
By similar reasoning we can prove another result*; the argument is
repeated in the example which follows ; if, for any set of values of the
integers ftlt ..., ftp, a1} ..., <xp, it is possible to obtain more than one set of
places nl} ... , np^l to satisfy the equations
then it is, of course, possible to obtain an infinite number of such sets. Let
oo i be the number of sets obtainable. Then fta. = q + 1 (mod. 2). And this
may be understood to include the general cases when (i) for an even value
of ft a, no solution of the congruence is possible (q = — 1), (ii), for an odd value
of fta, only a single solution is possible (q — 0).
As an example of the exceptional case here referred to, consider the hyperelliptic
surface ; and first suppose p = 3, the equation associated with the surface being
then we clearly have () = 28 = 2':'~1(2P— 1) ^-polynomials, each of the form (x - at) (x - ay),
\*/
of which the zeros are both of the second order. We have, however, also, a ^-polynomial,
of the form (.r-c)2, in which c is arbitrary, of which the zeros are both of the second
order ; denote these zeros by c and c ; then if |Qo o be a proper half-period
* Weber, Math. Ann. xin. p. 42.
185] FOR ZERO VALUES OF THE ARGUMENTS. 271
but, since, if e be any other place, the function (x-c)j(x-e) is a rational function, it
follows that (c, c) = (e, e)y and therefore that in the value just written for ^G^ a, c may
be replaced by e, and therefore, regarded as quite arbitrary. By the result (F), the
function Q(u) vanishes when u is replaced by %Qp a, and therefore e (^''-^fl^ J, which
is equal to Q(vx> m — vc> mi — vc>m'1 — 'if' ™3), vanishes when x is at c; since c is arbitrary the
function e (vx> z - |li^ J vanishes identically in regard to x, for all positions of z. If the
function Q(vx'z + vXl' Zl-Afl^ 0) vanished identically, it would, by Prop. VI., be possible,
in the equation
to choose both zl and z2 arbitrarily. As this is not the case, it follows, by Prop. XIV.
below, that the function G(?t + £Qo a), and its first, but not its second differential
coefficients, vanish for u = 0. Hence \Q^ a is an even half-period. (See the tables for
the hyperelliptic case, given in the next chapter, §§ 204, 205.)
There is therefore, in the hyperelliptic case in which p = 3, one even theta function
which vanishes for zero values of the argument.
In any hyperelliptic case in which p is odd, the equation associated with the surface
being
y1 = (x-av} ...... 0»-«2P + 2)
(^-polynomials with double zeros are given by
(i) the ( ] polynomials such as (x -04) ...... (x-ap_l). As there is no arbitrary
place involved, the q of the theorem enunciated (§ 185) is zero, and the half- period given by
the equation
where nfi ..., n\-i are the zeros of the (^-polynomial under consideration, is consequently
odd.
/2» + 2\
(ii) the ( 2 j polynomials such as (x — at) ...... (x-ap_3} (x-c)2, wherein c is
arbitrary. Here <? = ! and /3a = 0 (mod. 2).
(2v)_|_2\
r j polynomials such as (x-a^) ...... (x — a,, _ 6) (x — cf (x — e)2, for which
5 = 2, |3a= 1 (mod. 2) ; and so on. And, finally,
the single polynomial of the form (x-c^ ...... (.r-Cjj_j)2, in which all of clt ..., Cp_i
~iT ~2
are arbitrary ; in this case q=--~- , /3a=*-^- (mod. 2).
2i Jt
On the whole there arise
^-polynomials corresponding to odd half- periods, according as p=l or 3 (mod. 4).
Now in fact, when p= 1 (mod. 4)
4*)+ ......
272 EXAMPLE OF THE HYPERELLIPTIC CASE. [185
is equal to
while, when^> = 3 (mod. 4)
is equal to £ (^ + 2 - 2* + 2 cos ^^ TT} , and therefore, also to 2" ~ J (2" - 1 ).
Thus all the odd half-periods are accounted for. And there are
even half-periods which reduce the theta function to zero. This number is equal to
namely to 2P~1(2P + l)-( P+ J. This is the number of even theta functions which
vanish for zero values of the argument. It is easy to see that the same number is
obtained when p is even. For instance when jo = 4, there are 10 even theta functions
which vanish for zero values of the argument. They correspond to the 10 ^-polynomials
of the form (x - c)2 (x - at), wherein c is arbitrary, and a^ is one of the 10 branch places.
There are therefore ( ^ + ) even theta functions which do not vanish for zero values of
\ P J
the argument.
In regard to the places m1, ..., mp in the hyperelliptic case the following remark may
conveniently be made here. Suppose the place TO taken at the branch place a2p + 2 > using
the geometrical rule given in § 183, we may take for the polynomial A, of grade /*, the
polynomial #-a2p + 2, of grade 1; its remaining ?i/i-2, =0, zeros, give no conditions for
the polynomial ^ of grade (n- l)<r+»i-3 + /n, = (2- 1) p + 2-3 + 1, =p. Since o- + 1, the
dimension of y, is p+l, the only possible form for ^ is that of an integral polynomial
in x of order p. This is to be chosen so that its 2jo zeros consist of p repeated zeros.
When p = 3, for example, it must, therefore, be of one of the forms (x - a^ (x - a;} (x - a*},
(x-<ti} (x-cf, where c is arbitrary. It will be seen in the next chapter that the former
is the proper form.
186. Another matter* which connects the present theory with a subject afterwards
(Chap. XIII.) dealt with may be referred to here. Let £G be a half-period such that
the congruence
\Q. = vm>" m-tf" '"' - ...... -ir2"-1- m"-1
can be satisfied by oc« coresidual sets of places zlt ..., zp-l (as in Proposition VI.). Then
we have
(m2, z^, ...,22p_1) = (m18, ...,mp2),
so that (Prop. IX.) zlt ..., zp_lt each repeated, are the zeros of a ^-polynomial ; denote
this polynomial by 0. If zj, ..., zfr>_l be another set, which, repeated, are the zeros of a
0-polynomial <£', and are such that
Cf. Weber, Math. Annul, xni. p. 35; Noether, Math. Anna!, xvn. 263.
186] RESULTING RELATIONS CONNECTING <£- POLYNOMIALS.
then we have
273
0 = 2vmp' m-vz" m> - v*1'' m' - ...... -
so that SI,...,ZP-I, Zi, ..., «'p_, are the zeros of a ^-polynomial ; denote this polynomial
by iff.
The rational functions >//•/<£, </>'/^ have the same poles, the places zn ..., zp_lt and
the same zeros, the places zt', ...,2v_r Therefore, absorbing a constant multiplier in ^,
we have
^ = W, and 074> =
and thus the function V0'/<£ may be regarded as a rational function if a proper sign
be always attached. The function has zlt ..., zp_l for poles and z^, ..., z'p_1 for zeros.
Conversely any rational function having zlt ..., zp_l for poles can be written in this form.
For if Zj", ..., z"p-i be the zeros of such a function, we have
vz>"'z> + ......
and therefore, by the first equation of this §, also
thus q of the zeros can be taken arbitrarily ; and if $ be any ^-polynomial whose zeros
Cu ••• > fp-i are all of the second order, and such that
we can put
where fa, ..., 0, are particular polynomials such as </>' or *, andX, X1; ..., X, are constants.
In other words, corresponding to the GO « sets of solutions of the original equation of this
§, we have an equation of the form
wherein proper signs are to be attached to the ratios of any two of the square roots, and
any two of the q + l polynomials $, ^lt ...,$„ are such that their product is the square of
a 0-polynomial. There are therefore fa (q + l) linearly independent quadratic relations
connecting the ^-polynomials. (Cf. Chap. VI. §§ 110—112.)
For example in the hyperelliptic case in which p = 3, the vanishing of an even theta
function corresponds to the existence of a ^-polynomial * = (#-c)2, such that
where 0^3, =(*)*, =$*.
Ex. i. Prove, for j» = 3, that if an even theta function vanishes for zero values of the
arguments the surface is necessarily hyperelliptic.
Ex. ii. Prove, for jo = 4, that if two even theta functions vanish for zero values of the
arguments the surface is necessarily hyperelliptic ; so that, then, eight other even theta
functions also vanish for zero values of the arguments. The number, 2, of conditions thus
necessary for the fundamental constants of the surface, in order that it be hyperelliptic, is
the same as the difference, 9-7, between the number, 3p-3, of constants in the general
surface of deficiency 4, and the number, 2/>-l, of constants in the general hyperelliptic
surface of deficiency 4.
B- 18
274 INTEGRALS OF THE THIRD KIND [187
187. (XII.) If r denote any arguments such that @ (r) = 0, and such
that ®(vx'z + r) does not vanish identically for all positions of x and z,
the Riemann normal integral of the third kind can be expressed in the form
©(^>0+r)_|'
For consider the function of x given by
+ r) © (if' P + r)
" ~
(a) it is single-valued on the Riemann surface dissected by the a and b
period loops ;
(/3) it does not vanish or become infinite, for the zeros of ® (vx' z + r),
other than z, do not depend upon z (by Proposition IV.);
(7) it is unaffected by a circuit of any one of the period loops. At
a loop at it has clearly (Equation B, § 175) the factor unity ; at a loop
bi it has the factor
e l . e
which is also unity. Thus the function is single-valued on the undissected
surface ;
(8) thus the function is independent of x ; and hence equal to the value
it has when the place x is at zy namely 1.
A particular case is obtained by taking
where z1} ..., zp-i are any places such that ®(vx>z + r) does not vanish
identically. Then by the result (F) the function ® (r) vanishes.
Hence we have
Another particular case, of great importance, is obtained by taking
r = ^ftjt, tf, k, k' denoting respectively p integers k1} ...,kp, h', ..., kp', such
that kk' is odd, the assumption being made that the equations
188] EXPRESSED BY THETA FUNCTIONS. 275
are not satisfied by more than one set of places £,, ..., £p_j (cf. Props. III., V.).
Then the function © (ifK>z + ^Qk,k) does not vanish identically, and we have
(XIII.) Suppose k equal to or less than p ; consider the function given
by the product of
g-n^-n^- -n*;^
and
(H) (i&i m 9)°" wi -^ — V^k' mk -t- ^^ / @ ^D2' "* li"1 ' wl' — Wa*' *"* -4- T}
wherein r denotes arguments given by
and each of the sets a1} ..., a^, 7^+1, •••,%,&, •••, 0k, 7*+i> •••» 7p ^s sucn
that the functions involved do not vanish identically in regard to x.
This function is single-valued on the dissected Riemann surface, does not
become infinite or zero, and, for example, at the period loop 6; it has the factor
eL, where
L, = - 2m (v°> ' P' + ...... + v°>;> h) - 2-rn (v*> m - -va" ™- -
+ 2-Trt (va> »» - 1^1
is zero. Thus the function has the constant value, unity, which it has when
x is at z. Therefore
na;(3 x,z _,
+ ' ' ' + ~ °
the places 7i+1, ...,<yp being arbitrarily chosen so that a1} ...,«£,
are not zeros of a ^-polynomial, and /3lt ..., ^8^, 7^+1, . .., yp are not zeros of a
^-polynomial.
Thus, when k = p, we have the expression of the function considered in
§171, Chap. IX. in terms of theta functions. For the case where oti, ..., a*
are the zeros of a ^-polynomial, cf. Prop. XV. Cor. iii
188. (XIV.) We return now to the consideration of the identical vanishing
of the ® function. We have proved (Prop. VII.), that if ®(vx"z>+ ......
+ tfB«'z<« + r) be identically zero for all positions of xl> ,..,xq,z-i, ...,zq, but
(5) (rx, z + vx, , z, + ...... + 7,a-,, z,, + r} ^e not i(jentically zero for all positions of
18—2
276 INVESTIGATION [188
x and z, then there exist oo « sets of places £,..., £p_i, and oo 9 sets of places
£, ...,&>_!, such that
and
_ j- = <)finp, in _ yfi , m, _ ^ _ ^ _ •yfp-i. *»j»-it
Now, if in the equation ® (^ > *> + ...... + t^ *« + r) = 0, we make xq
approach to and coincide with zq, we obtain
t 6/ (tf*'> *> + ...... + v*'-1' z<-' + r) ^ (s9) = 0,
i=l
•
wherein ©/ (it) is put for r— ® (u), H; (a;) for 2-Trt Dxvf ", a being arbitrary ;
(7Mj
and this equation holds for all positions of ac1, z1} ..., #9_a, ^9_i. Since, how
ever, the quantities fl] (zq), ... , flg (^q) cannot be connected by any linear
equation whose coefficients are independent of zq, we can thence infer that
the first differential coefficients of ® (u) vanish identically when u is of the
form^" z> + ...... +0*9-1.29-1 + T». It follows then in the same way that the
second differential coefficients of @ (u) vanish identically when u has the
form vXi> 2>+ ...... + vxi-°-> z<>-2 + r ; in particular all the first and second differ
ential coefficients vanish when u = r. Proceeding thus we finally infer that
® (w) and all its differential coefficients up to and including those of the <jth
order vanish when u = r.
We proceed now to shew conversely that when © (u) and all its differential
coefficients up to and including those of the <?th order, vanish for u = r,
then © (vXl ' z> 4- ...... +^9.29+r) vanishes identically for all positions of
#1, ^i, ^2,^2, ...,xq,Zq. By what has just been shewn © (v** z + v*"*! + ......
+ -y*,, zq + rj wiH not vanish identically unless the differential coefficients of
the (q + l)th order also vanish.
We begin with the case q=l. Suppose that ® (M), ©/ (u), ... , Sp' (u), all
vanish for u = r ; we are to prove that ® (#*• z + r) vanishes identically for all
positions of x and z.
Let e, f be such arguments that @(e)=0, ® (/)=(), but such that
©/ (e) are not all zero and ©/(/) are not all zero, and therefore ® (vx>
(J0 not vanish identically; consider the function
@ (e + if6' z) @ (e - tf> z)
firstly, it is rational in x and z ; for, considered as a function of x, it has,
at the period loop br, (Equation B, § 175) the factor
188] OF THE GENERAL PROBLEM 277
whose value is unity ; and a similar statement holds when the expression is
considered as a function of z, for the expression is immediately seen to be
symmetrical in x and z ; secondly, regarded as a function of x, the expression
has 2 (p — 1) zeros, and the same number of poles, and these (Prop. IV.)
are independent of z. Similarly as a function of z it has 2 (p — 1) zeros and
poles, independent of a; ; therefore the expression can be written in the form
F(x}F(z}, where F(x) denotes the definite rational function having the
proper zeros and poles, multiplied by a suitable constant factor, and F (z) is
the same rational function of z.
Putting, then, x to coincide with z, and extracting a square root, we infer
p
where flt (a?) = 2m Dxv*'a, for a arbitrary, is the differential coefficient of an
integral of the first kind ; thence we have
In this equation suppose that e approaches indefinitely near to r, for which
®(r) = 0, ®/(r) = 0. Then the right hand becomes infinitesimal, inde
pendently of x and z. Therefore also the left hand becomes infinitesimal
independently of x and z ; and hence 0 (vx' z + r) vanishes identically, for
all positions of as and z.
We have thus proved the case of our general theorem in which q = l.
The theorem is to be inferred for higher values of q by proving that if the
function ® (vx>< '• + ...... + t^»->' *-> + r) vanish identically for all positions of
#!, 2j, ... , #,„_!, 2m_i, and also the differential coefficients of ®(w), of order
m, vanish for it = r, then the function ®(^i.«i + ...... +^», «« + r) vanishes
identically. For instance if this were proved, it would follow, putting m = 2,
from what we have just proved, that also ®(vx>< ^ + if*'** + r) vanished
identically, and so on.
As before let/ be such that ® (/) = 0, but all of ©/ (/) are not zero ; so
that ® («*. z +/) does not vanish identically in regard to x and z. Let
e be such that ®(v«..^ + ...... + w*»-i,«w-i+e) vanishes identically for all
positions of #,, *„ iV.ViUi Zm-i , but such that the differential coefficients of
®(M) of the first order do not vanish identically for u = vxi>zi+... + •y*»«-i,«'«-i -j. e-
so that the function @ (t^i. ^ + ...... + &m, ^ + e) does not vanish identically.'
Consider the product of the expressions
e)
_
nn® (w«x. v +/) 0 (t^x. v -/)
278 OF THE IDENTICAL VANISHING [188
wherein h, k in the numerator denote in turn every pair of the numbers
1,2, ... , m, so that the numerator contains 4 . \m (m — 1) + 2 = 2 (m2 — m + 1)
theta functions, and X, /-t in the denominator are each to take all the values
1, 2, ..., m, so that there are 2m2 theta functions in the denominator.
Firstly, this product is a rational function of each of the 2m places
a?!, Zi, ..., xm, zm. Consider for instance ccl; it is clear that if the product
be rational in xlt it will be entirely rational. As a function of xl} the
product has at the period loop br a factor e~2iriK where
and this expression is identically zero.
Secondly, considering the product as a rational function of x1} the
denominator is zero to the second order when x± coincides with any one of
the m places zlt ..., zm, and is otherwise zero at 2m (p — 1) places depending
on / only ; of these latter places 2 (m — 1) (p— 1) are also zeros of the
factors H'0 («**'**+/)©(«**•**--/); there are then 2(jp-l) poles of the
function which depend on / only. The factors II'® (tf"*« ** +/) ©(«"*« ** -/)
have also the zeros #2> •••> #»»> each of the second order. The factors
® (tfii ,«i + ...+ tf*. , zm + e)@ (t^i , zi + . . . + if* , z». _ e) have, by the hypothesis
as to e, the zeros zlt z2} ..., zm, each of the second order, as well as 2 (p — m)
other zeros depending on e only. On the whole then, regarded as a function
of ac1 , the product has
for zeros, 2(p — m) zeros depending on e, as well as the zeros #2, ..., xmy
each of the second order,
for poles, 2 (p — 1) poles depending on/;
the function is thus of order 2(p — 1); and it is determined, save for a
factor independent of xly by the assignation of its zeros and poles. It is
to be noticed that these do not depend on z1} z2, ..., zm.
It is easy now to see that the product, regarded as a function of zl}
depends on z2, ..., zm, e,f in just the same way as, regarded as a function
of x1} it depends on cc2, ..., xm, e,f.
The expression is therefore of the form F(xlt #2> •••> ®m) F (z\> 2-2, •••> zm\
wherein F denotes a rational function of all the variables involved.
The form of F can be determined by supposing xlt ...,xm to approach
indefinitely near to z1} ..., zm respectively; then we obtain
where tm is the infinitesimal for the neighbourhood of the place zm,
0.' ^, ,Zl+ ...... + vX,n-l, Z«
188] OF THE THETA FUNCTION. 279
where tm-i is the infinitesimal for the neighbourhood of the place 2m_i, and
so on, and eventually,
\47rl) fm = i t'i=l
Similarly
where /i, A; refers to all pairs of different numbers from among 1, 2, ..., m.
Therefore, dividing by a factor
which is common to numerator and denominator, and taking the square root,
we have
i. i e'^, j,,. ... iw (<o HI (^on,^. ..««(*»)
, t>»
i ' ' ' ' zm) ~
On the whole therefore we have the equation
!• z' + + w*m'ZBl - e)
* - IT® * . zt + @
nn@(/A>
•vEr/'/v. r f>\ty(? ? o\
•K ^! , . . . , J/m , K) X \^i, . • • , 4m > K)
where
^(^,...,^,6)= 2 ... @'il)r-2 ..... ^(ejfl^fo)... flfM(ff,w).
l-m = l H = l
Suppose now that ef is made to approach to r; ; then the conditions we
have imposed for e are satisfied, and there is added the further condition
that the differential coefficients of order m, O',-^ ,-2) ...,,•„>, also vanish. Hence
it follows that ® (^ <z> + ...... + 1^«. z-» + ?•) vanishes identically.
The whole theorem enunciated is thus demonstrated.
280 RESULTING EXPRESSION [188
(XV.) The remarkable investigation of Prop. XIV. is due to Riemann ;
it is worth while to give a separate statement of one of the results obtained.
Using q instead of m — 1, we have proved that if the equations
Q = vmp> m — i1' m' — . . . . — I;P
are satisfied by oc 1 sets of places fi, ..., ^_1( so that also the equations
— e = v™*1 ' m — v^ > ' rrti — ...... — ?;£ p
are satisfied by x« sets of places £, ..., f^, then their exists a rational
function, which has (i) for poles, the 2(p — 1) places tl} ..., tp,l} zlt ..., zp^,
which satisfy the equations
f=
f being supposed such that these equations have one and only one set of
solutions, and has (ii) for zeros, the arbitrary places xlt ...,acq, each of the
second order, together with 2(p- 1 -q) places £9+1, ..., Zp_l} j-q+1, ..., g^,
satisfying the equations
Q = Vmp' m — iF1' m' — ...... — •y2'9' 1Hq — l)£<i+l> m4+i _ .. .. _ yfp-i» mp-i
— e = vm*' m — V^1' TOI — . . . . — VXq> inq — ifig+1> niq+l _ . . . . _ •yfp-1' *»?-!
and the function can be given in the form
¥(#!, xz, ..., acq, x,e) + ® (a;,/),
the notation being that employed at the conclusion of Proposition (XIV.).
The expressions M', $ occurring here have the zeros of certain (^-polynomials,
to which they are proportional.
Corollary i. If we take p — 1 places £, ..., ^p_1) so situated that only
one ^-polynomial vanishes in all of them, and define e by the equations
e =
there will be no other set £i, ..., ^,_1( satisfying these equations, or 5=0.
If fi> •••> Zp-i be the remaining zeros of the ^-polynomial which vanishes in
£i, ... , %p-i, we have (Prop. IX.)
(m\ 5i, ..., ^_!, £, ..., 1^0 = (wh8, .... 7?i/),
and therefore
Similarly if tlf . . . , <p_! be arbitrary places which are the zeros of only one
^-polynomial, we can put
f ' — <yWiy), m _ n,tt , m, _ _ OI^P-II Wp-i
— f= vm'" '" — vz> > m< — ...... — v*"-1- '"p-1.
188] OF RATIONAL FUNCTIONS 281
Then the rational function having tit . .., £p_i, z^, ..., Zp-i for poles, and
(Ti. •••. KP-I, &> ••-. &-i for zeroy is given by ^(^ «) -5- <& (#> /)• Thus tne
^-polynomial which vanishes in %\> • ••> £p-i> £i> •••> £p-i is given by
P
2 ®/ (vm»' m — v^'m> — ...... — Vs"-1' mp~l) fa (x),
1=1
where fa (a), ...,<}>p(x) are the ^-polynomials occurring in the differential
coefficients of Riemann's normal integrals of the first kind.
Hence if n^ ..., np_! be places which, repeated, are all the zeros of a
^-polynomial, the form of this polynomial is known. Since, then, we have
(Prop. XI. p. 269)
we can write this polynomial
|fl being an odd half-period.
If another ^-polynomial than this one vanished in nlt ..., np_lt there
would be other places n^, ..., rip_1} such that
and therefore (Prop. VI.) the function B(^»*.+ ^(l) would vanish identi
cally; in that case (Prop. XIV. p. 276) the coefficients ®/(£H) would vanish.
We can express the 0-polynomial in terms of any integrals of the
first kind; if Vi '",..., Vp' " be any linearly independent integrals of
the first kind, expressible in terms of the Riemann normal integrals
v*' , ..., Vp m by linear equations of the form
*• m -\ -irx> m . , -v irx< m /-to
Vi =X;)1F1 + ...... +\,pVp , (l = l, 2, ...,/>),
and the function ® (u) be regarded as a function of Ul} . . . , Up given by
w» = Xi>1 t/i4 ...... + \iif Up, (i=l, 2, ...,p),
and, so regarded, be written ^ ( U), the ^-polynomial which has zeros of the
second order at nlt ..., wp_i can be written
'm
where ^ (a;), . . . , -^ («) are the ^-polynomials corresponding to V?
V*'m, and ^H denotes a set of simultaneous half-periods of the integrals
FI ' '", ..., Vfl'm. If ^n stand for p quantities of which a general one is
282 OF SPECIAL KIND. [188
and (»,., s, w'r, g be 2p- quantities given by
J- = 2\ j «i, 8 + 2X;, 2 ft)2i g + ...... + 2X/( p Op, ,, (t, 5 = 1, 2, . . . , p),
Tt-, g = 2\i, ] o)'^ g -I- 2X,-, 2 a/-, g + ...... + 2\it p a>'p> 8,
where, in the first equation, we are to take 1 or 0 according as i = s or i^s,
then £fl will stand for p quantities of which one is
Aa®*, i + ...... + kpcot, p + ki(o'it i + ...... + kp'o)'it p, (i = I, 2, . . . , p).
For example when the fundamental Riemann surface is that whose
equation may be interpreted as the equation of a plane quartic curve, every
double tangent is associated with an odd half-period and its equation may
be put into the form
*&/ (in) + jfo' (f ii) 4- *; (in) = o.
Corollary ii. If the equations
e = vm>" m — if1' Wi — v^2 ' m* _ . . — v^p~l> mi>-1
can be satisfied with an arbitrary position of x^ and suitable positions of
£,, ..., %p-i, and therefore, also, the equations
— e =
can be satisfied, then a ^-polynomial vanishing at x1 to the second order, and
otherwise vanishing in £,, ..., £p_!, ^2, •••, fp-i, is given by
Ex. In the case of a plane quintic curve having two double points, this gives us the
equation of the straight lines joining these double points to an arbitrary point x1, of the
curve.
Corollary iii. We have seen (Chap. VI. § 98) that any rational function
of which the multiplicity (q) is greater than the excess of the order of the
function over the deficiency of the surface, say, q = Q—p + r + I, can be
expressed as the quotient of two ^-polynomials. If the function have
£i, •••> & f°r zeros, and £, ..., J?Q for poles, and the common zeros of the
^-polynomials expressing the function be zlt ..., ZR, where R=2p-2-Q,
the function is in fact expressed by
where (cf. § 93, Chap. VI.)
7?ln * 7ft £1 » tJl-t
0 n\ ir* nt A' ^ <1
/. 7)(D. Ill 2,, HJj *B_T,
f=vp — v — — v
189] GENERALIZED THETA FUNCTION. 283
189. Before concluding this chapter it is convenient to introduce a
slightly more general function * than that so far considered ; we denote by
^ (u ; q, q'), or by S- (u, q), the function
& (u ; q, q') = 2eaw2+2AM<w+9'>+&(n+9')2+2"r9(ri+5'>,
wherein the summation extends to all positive and negative integer values of
the p integers wn ...,np,a is any symmetrical matrix whatever of p rows and
columns, h is any matrix whatever of p rows and columns, in general not
symmetrical, b is any symmetrical matrix whatever of p rows and columns,
such that the real part of the quadratic form bm2 is necessarily negative
for all real values of the quantities m1, ..., mp, other than zero, and q, q
denote two sets, each of p constant quantities, which constitute the character
istic of the function. In the most general case the matrix b depends on
?p(p+ 1) independent constants ; if however we put iirr for b, r being the
symmetrical matrix hitherto used, depending only on 3p - 3 constants, and
denote the p quantities hu by U, we shall obtain
*(M; q, q') = eau* ® (U ; q, q').
We make consistent use of the notation of matrices (see Appendix ii.).
If u denote a row (or column) letter of p elements, and h denote any matrix
of p rows and columns, then hu is a row letter ; we shall generally write
huv for hu.v; and we have huv = hvu, where h is the matrix obtained from
h by transposition of rows and columns. Further if k be any matrix of p rows
and columns, hu . kv = hkvu = khuv. For the present every matrix denoted by
a single letter is a square matrix of p rows and columns.
Now let o>, w', r), T/ be any such matrices, and P, P' be row letters of
elements Pl} ..., Pp, P/, ..., Pp. Then, by the sum of the two row letters
a>P + oa'P' we denote a row letter consisting of p elements, each being the
sum of an element of o>P with the corresponding element of &>'P'. This
row letter, with every element multiplied by 2, will be denoted by flp,
so that
in a similar way we define a row letter of p elements by the equation
HP = 2r)P + 27/P' ;
then u + flp will denote a row letter of p elements, like u.
The equation we desire to prove, subject to proper relations connecting
&), a/, 77, 77', is the following,
* (u + ft,,, q) = effp(«+sn,)-Wip/'+2« (Pq'-p-q) e-z«iPV' § (W) p + 9)} (L^
which is a generalization of some of the fundamental equations given for
© (u).
* Schottky, Abrias einer Theoric der Abehchen Functionen von drei Variabeln, Leipzig, 1880.
The introduction of the matrix notation is suggested by Cayley, Math. Aniuil. (xvn.), p. 115.
284 THE FUNDAMENTAL EQUATIONS. [189
In order that this equation may hold it is sufficient that the terms on the
two sides of the equation, which contain the same values of the summation
letters %, ..., np, should be equal ; this will be so if
a (u + dp)2 + 2h(u + flp) (n + q') + b(n + q'f + 2iriq (n + q)
= HP(u + £flp) - TriPP' - ZTriP'q + an" + 2hu (n + q' + P') + b (n + q' + PJ
picking out in this conditional equation respectively the terms involving
squares, first powers, and zero powers of n1} ..., np, we require
6 = 6,
h (u + tip) + bq + Triq = hu+b (q + P') + Tri(P + q),
and
a (u + Hp)3 + 2/i (u + Hp) q + bq'2 + 2-rriqq' = HP (u + £HP) - TriPP' - liriP'q
+ cm2 + 2/m (q' + P') + 6 (q + P')2 + Ziri (P + q) (q' + P').
190. In working out these conditions it will be convenient at first to
neglect the fact that a and 6 are symmetrical matrices, in order to see how
far it is necessary.
The second of these conditions gives
MIP = TriP + 6P',
and therefore gives the two conditions hw = ^TTI, hw = ^5, whereby o>, &>'
are determined in terms of the matrices h, b. In particular when h = iri
and 6 = i7TT, as in the case of the function ®(V), we have 2a> = l, 2&>' = T,
namely 2o>, 2o>' are the matrices of the periods of the Riemann normal
integrals of the first kind, respectively at the first kind, and at the second
kind of period loops.
The third condition gives
ZauSlp + aO2P + 2/tOp^' = HP (u + ^flp)
- iriPP' - ZiriP'q + ZhuP' + b (2q'P' + P'2) + 2™ (qP' + Pq + PF),
that is
-Hp- IhP1) u + (aflp -$HP) flp - TriPP' - 6P'2
+ 2 (AnP - triP - 6P') q' = 0 ;
in order that this may be satisfied for all values of n1} ..., UP) we must have,
referring to the equation already obtained from the second condition,
and
6P') P' ;
from the first of these, by the equation already obtained, we have
kttpP' = (-rriP + bP') P ;
190] THE RELATIONS FOR THE PERIODS. 285
subtracting this from the second equation, there results
and in order that this may hold independently of the values assigned to
P, P' it is necessary that d = a,b=b', when this is so, these two equations
give, in addition to the one already obtained, only the equation
leading to
77 = 2a&>, V}' = 2aw' — 2h,
which express the matrices 77 and r[ in terms of the matrices a and h. These
equations, with
or
hca = \irit ha*' = ^b,
are all the conditions necessary, and they are clearly sufficient. When they
are satisfied we have
- q + P), (L),
where
XP (w) = HP (u + | Hp) - -rriPP.
Ex. Weierstrass's function cru is given by
where A is a certain constant.
The equations obtained express the 4>p- elements of the matrices &>, to', 77, 77'
in terms of the pz + p (p 4- 1 ) quantities occurring in the matrices a,h,b\
there must therefore be 2p2 — p relations connecting the quantities in o>, «',
v), 77'. The equations are in fact of precisely the same form as those already
obtained in § 140, Chap. VII., equation (A), and precisely as in § 141 it
follows that the necessary relations connecting G>, &>', 77, 77' may be expressed
by either of the equations (B), (C) of § 140. Using the notation of matrices
in greater detail we may express these relations in a still further way.
For
- hP) 0Q - (ang - hF) ftp
= hflp . Q' - h£lQ . P'
= (viP + bP) $ - (iriQ + bQ') P,
so that
- PQ) ;
this relation includes all the 2p2 — p necessary relations ; for it gives
(rjP + rj'P) (a>Q + u'Qf) - (r,Q + rj'Q') (a>P + to'P) = ^-rri (PQ' - PQ),
286 CASE WHEN THE FUNCTION IS ODD OR EVEN. [190
or (using the matrix relation already quoted in the form hu.kv= 7ikvu =
(wrj - tjto) PQ -f <W - ^o>') P'Q + (W'T; - rj'w) PQ' + (m'rj' - ijV) P'Q'
and expressing that this equation holds for all values of P, Q, P', Q', we
obtain the Weierstrassian equations ((B) § 140).
Similarly the Eiemann equations ((C) § 140) are all expressed by
') - (2a>P + 2rjQ) (2«'P' + 2ij'Q/)= 27rt (PQ' - P'Q).
Ex. i. If we substitute for the variables u in the ^ function linear functions of any p
new variables v, with non-vanishing determinant of transformation, and LP be formed from
the new form of the ^ function, regarded as a function of v, just as HP was formed from
the original function, prove that LPv = HPu, and that XP (u) remains unaltered.
Ex. ii. Prove that
XP (u + n.v) + X.v (u) - 2TriM'P= Xfi (u + fl.v) + X.v (u} -
provided
The equation (L) is simplified when P, P' both consist of integers. For
if M, M' be rows of integers, it is easy (putting a new summation letter,
m, for n + M', in the exponent of the general term of ^ (u ; q + M, q' + M'),)
to verify that
$(u; q + M, q' + M') = &™M<t ^ (u ; q, q').
Therefore, if m, m' consist of integers, we find
S- (u + Slm,q) = eKm^ +***(«*' -m'q] ^ (Wj 3)}
and in particular
S(w + ftm) = eAm(M)Sr(*0>
where ^ (u) is written for ^r (u ; 0, 0). The reader will compare the equations
obtained at the beginning of this chapter, where a = 0, 77 = 0, vf = — 2m,
a, = £, a' =£T, flp = P + rP', HP=- 2-rriP', \P (u) = - MF (u + %P + %rP)
- TriPP'.
One equation, just used, deserves a separate statement ; we have
*(u; q + M) = e*"iMi' % (u ; q),
where M stands for a row of integers M1} ..., Mp, MJ, ..., Mp'.
191. Finally, to conclude these general explanations as to the function
^ (u), we may enquire in what cases ^ (u) can be an odd or even function.
When m, m' are rows of integers the general formula gives
^(-11 + flm, q) = &» (-») +2™ (?»<?' -*»'<?) ^ (- u, q) ;
192] INTRODUCTION OF THE £ FUNCTIONS.
hence when ^ (u, q) is odd, or is even, since \,n(— u) = \-m (u), we have
<\ /?/ _ O n} — /jA_m(M) + 2m'(mo'— m'q) «\ /». n\ .
,J \IV *I"Hl) <£) t/ rJ \U, »// ,
therefore, by equation (L),
j (u + i.im, q), = j (u — ±lm, q) . e^~" ,
while also, by the same equation,
Thus the expression
^SOT (u — i ft™) + X_m (u) — \m (u) + kin (mq — m'q)
must be an integral multiple of 2?™'. This is immediately seen to require
only that 2 (mq' — m'q — mm') be integral for all integral values of m, m'.
Hence the necessary and sufficient condition is that q and q' consist of half-
integers. In that case we prove as before that ^ (u, q) is odd or even
according as 4>qq' is an odd or even integer.
192. In what follows in the present chapter we consider only the case in
which b = ITTT, r being the matrix of the periods of Riemann's normal
integrals at the second kind of period loops. And if if a,..., wj a denote
any p linearly independent integrals of the first kind, such as used in §§ 138,
139, Chap. VII., the matrix h is here taken to be such that
7 , a , x, a
=hi>1ul + ...... + hitpup , (i-l, 2, ...,
so that h is as in § 139, and
• (u*> a, q) = eauz © 0*' « q),
where u = ux> a.
From the formula
S- (u + flm) = ea»(«+jo»)-««w»' ^ (w),
wherein m, m' denote rows of integers, we infer, using the abbreviation
a
£(w) = ^.logS-O)>
that
& (u + flm) - & (u) = 2 (rjit 1 m, + ...... +rjiiltmp + iifiilm1t+ ...... -i-^.^
particular cases of this formula are
f / (M, + 2a>lf ,., . . . , up + 2a,pt ,.) = £ (u) + 2^ r,
& (w, + 20)',, r, ..., up + 2a)'p! r) = Si CM) + 2i/i, r>
288 THE DIFFERENCE OF TWO f FUNCTIONS [192
Thus if us be the argument
where MI , ...,?4'a are any P linearly independent integrals of the first
kind, and the matrix a here used in the definition of & (u) be the same as
that previously used (Chap. VII. § 138) in the definition of the integral
L^ , so that the matrices ?;, rf will be the same in both cases, then it
follows that the periods of the expression
regarded as a function of x, are zero.
193. And in fact, when the matrix a is thus chosen, there exists the
equation
— £;• (ux> m — ux> ' m' — ...... - ux"> mt>) + & (ua> m — ux» Wi - ...... - ux"> m*)
TX. & . ^* ~ r / \ / \T
= L' +2 vr>i [(xr, x) - (xr, a)]
r=l
=- ,
wherein vr> i denotes the minor of the element /^- (xr) in the determinant
whose (r, i)th element is pi(xr), divided by this determinant itself; thus
vr,i depends on the places xlt ..., xp exactly as the quantity vr> { (Chap. VII.
§ 138) depends on the places c1} ..., cp.
For we have just remarked that the two sides of this equation regarded as
functions of x have the same periods ; the left-hand side is only infinite
at the places xlt ...,xp; if in Lfa, which does not depend on the places
GI, ..., Cp used in forming it (Chap. VII. § 138), we replace c1} ..., cp by
x1, ..., xp, it takes the form
-p,a;, a -^x, a _ . x, a x, a,
and becomes infinite only at the places acl} ...,acp. Hence the difference
of the two sides of the equation is a rational function with only p poles,
#j, ..., xp, having arbitrary positions. Such a function is a constant (Chap.
III. § 37, and Chap. VI.) ; and by putting x = a, we see that this constant is
zero.
194. It will be seen in the next chapter that in the hyperelliptic
case the equation of § 193 enables us to obtain a simple expression for
£i (ux> m —ux»m> — ...... — u^' mp) in terms of algebraical integrals and rational
functions only. In the general case we can also obtain such an expression* ;
* See Clebsch und Gordan, Abels. Functnen. p. 171, Thomae, Crelle, LXXI. (1870), p. 214,
Thomae, Crelle, ci. (1887), p. 326, Stahl, Crelle, cxi. (1893), p. 98, and, for a solution on different
lines, see the latter part of chapter XIV. of the present volume.
194] EXPRESSED BY ALGEBRAICAL INTEGRALS. 289
though not of very simple character (§ 196). In the course of deriving that
expression we give another proof of the equation of § 193.
The function of x given by & (%*•»«; |/3, |a) will have p zeros, unless
^ (ux> m + £ft0, a) vanish identically (§§ 179, 180) ; we suppose this is not the
case. Denote these zeros by m/, ..., mpf. Then (Prop. X. § 184) the function
^ (u*> m - «*• ' '«•' - ...... - U*P> 'V ; £/3, £a) will vanish when as coincides with
#1, a?2, ..., or xp. Determining mlt ... , mp so that
ump,mp' =
and supposing the exact value of the left-hand side to be £ flftt „ + £lk> h>
where k, h are integral, this function is equal to
%(u*,m_ux>,mi _ ...... _ uXp,mit _ in^a - nt>h ; ££. | a),
and this, by equation (L) is equal to
where u = ux> m — ux> > m> — ...... — uxi» mv — £lk h.
Therefore (§ 190) the expression
-W^.^/- ...... _w^,rV; /9,
is equal to
we may write this in the form
the expression is therefore equal to
;, m _ ^, mt _
, m _ ,, m, _
where
is equal to
or
-2a(U-V)(r-s),
that is
B- 19
290 DEDUCTION OF A FORMULA [194
which denotes
£ vv x, M *,, nr.
— 2, (2S2af,j«,- w; ).
T = 1 1 , j
Hence, by Prop. XIII. § 187, supposing that the matrix a, here used, is the
same as that used in § 138, Chap. VII., and denoting the canonical integral
TT a;, a ~ J} Ji x, a z, c
_ •/ > ^ /f -jy -i/
J-J.,y ^» A< — W ^W U.J- J tt-j« tfr^f j
r=ls=l
which has already occurred (page 194), by Rxz'>c , we have
195. From the formula
p x,p <bux'm-ux»
X, * <^Sux, m_ WMI, Wi — ... — ufp> mP
since
Xr, Mr *r.^ £ /V.^r*,*
-n-ar, /a = f x, /x + ^ ui Li '
t = l
we obtain
P D^.,^ P ^ •+*;*.*"', *(u*-m-U) /*(u»>™- U)
A *" ^-ir-l^ 10g^(^--- J70)/ ^(^'Wl- Z/o)
where
and therefore
U-U0=% u*"»*.
r = l
Hence, differentiating,
5 |^f [(*,, «) - (av, A*)] + Lr - - Si (**• m - U] 4- C, (^ - - IT),
r=ldC'i
where
but, from
where rfa?1; ..., rf^ denote the infinitesimals at xlt ..., xp, we obtain
9a;r _ dxr
dwr^^'
thus
P
- ?i (w*- m - U) + & (M* m - t^) = 4' m + 2 J>r, { [(xr , x) - fa, /*)
r=l
which is the equation of § 193.
196] TO EXPRESS A SINGLE £ FUNCTION. 291
196. From the equation
differentiating in regard to a;, we obtain an equation which we write in
the form
F? ^=ft (x) [f , («*. » - 17)- f r (M*. » - £T0)],
r=l r=l
where U=
Thus, if we take for ft , . . . , ft, places determined from a; just as mlt ..., mp
are determined from m, so that
(m, ft, ...,ft) = (x, TOJ, ..., mp),
the arguments ux> m- U0 will be = 0 ; as the odd function £V (u) vanishes for
zero values of the argument, we therefore have (§ 192), writing Hp for the
exact value of ux> m - U0)
r=l
P
C) £V (lix' m — llz> ' w> — ... — UZP > mr> — ftp)
r=l
V / \r i z
= — Z, ft- (X) %r (Wz» ^ + . . . + MZP' **»),
r=l
If in this equation we put x at TO we derive
p
1Tfzii ™i . |^ Jp^P' 'm^ V / \ y / Z Wl i Zm\ /TVT\
where ^1; ..., zp are arbitrary.
If however we put x in turn at p independent places clt ..., cp, and
denote the places determined from a, as ml} ..., TOP are determined from
m, by c^ !, ..., Citp, so that
(\ / \
/• • fjfYl nffl i — I /yi") /^ . /* . i
f 1 J »»H ) • • • y IIVipI I f/(/j (y-^ J j . , . ; C/^ ?>/»
we obtain p equations of the form
Suppose then that x, xl, ..., xp are arbitrary independent places; for
z-i, ..., zp put the places xijly ..., xiif determined by the congruence
(x, xit j, . . . , xit p) = (d, xl} ...,xp)\
then, if ftQ denote a certain period, — uXi' 1J Ci)i — ... — uXi'*» C<)P is equal to
HQ + ux> m - uxi ' mi - ...... - ux»> m", and we have
j»*<. i - ci, i , , JJT *i, P, cf, P _ ^ , } L,
t'i ' * fc ~ — A4/' X6'/' br
xp,
19—2
292 INTRODUCTION OF THE $> FUNCTIONS; [196
therefore
X> «.
-u -...- =
V . rvxr,l
= 2, vr> i [t Cr
where vr> i is the minor of ^i (cr) in the determinant whose (r, s)th element is
fa (0^, divided by the determinant itself.
In particular, when the differential coefficients ^ (x), . . . , pp (x) are those
r x
already denoted (§ 121, Chap. VII.) by wl (x), ...,wp (x}, and Vf a = I wi(x)dtx,
J a
and the paths of integration are properly taken, we have*
*'
( V*' m - Vl> l
197. A further result should be given. Let #, x1} ..., xp be fixed
places. Take a variable place z, and thereby determine places ziy ...,zp,
functions of z, such that
(x, zlt ..., Zp) = (z, «i, .... a?P).
Then from the formula
z, a
i r/ * * *.-, Zg
+2 vs> i [(zs, z) - (zs, a)] -=- ,
=i
=-
wherein vs, i is formed with zl} . . . , zp , we have, by differentiating in regard
to z and denoting — — ^ (M) by g)^ j (w),
3 = 1
' dt
where U=uz' m — uz" m> — _u2*'mp, U = ua< m — MZ» w- — — ifr- mv.
In this equation a is arbitrary. Let it now be put to coincide with z ;
hence
* This form is used by Noether, Math. Annal. xxxvn. (1890), p. 488.
197] THEIR ALGEBRAICAL EXPRESSION. 293
Therefore
ft, (IT)
f=l
= D; I ^ (A) /if' B + I «« (A;) [(*„ ^) - (^ a)] 1 ,
where A' means a differentiation taking no account of the fact that zl} ..., zp
are functions of z,
U=i
, « ; k, Z,, . . . , Zp)l,
in which form the expression is algebraically calculable when the integrals
L*'a are known (Chap. VII. § 138),
= D'z \ rj a - ty (z, a ; k, zlt . . . , zp) - 22Sar> , /*., (/.;) w
where c is an arbitrary place ; and this (cf. Ex. iv. § 125)
p P
= - W(z; k, zlf ..., fp)-2 S S
r=l*=l
If now
so that
= Uk> m - Uk> ' 7ni - ...... — U*l»
and
(a?, ^, ..., fp) = (^, a?lf ..., a^),
(ar, A-j, ..., kp)= (k, ar1} ..., ajp),
then the formula is
1 3 r=l«-l
by Ex. iv. § 125.
= W(k- z,k1,...ikp) + 2 I I ar,
r=ls=l
294 EXAMPLES. [197
By the congruences
Uz»' ^ = Uz> x
the places zl} ...,zp are algebraically determinable from the places x, xl , . . ., xp, 2,
and therefore the function W ' (z; k, zl} ..., zp) can be expressed by x, xlt ...,
xv, k, z only. In fact we have
/ \ r\
The interest of the formula lies in the fact that the left-hand side is a
multiply periodic function of the arguments U1) ..., Up.
A particular way of expressing the right-hand side in terms of x, xlt ..., xp, 2, k is to
put down %p(p + l} linearly independent particular cases of this equation, in which the
right-hand side contains only x, xlt ..., xp, z, k, and then to solve for the \p(p + l)
quantities j^y. Since ^ (z, a ; k, z1} ..., zp) vanishes when k = zp, we clearly have, as one
particular case,
22J0- • (uz> m— uZl' mi — — uss>'''m''']u-(z}iLt(z} = DD Rz' a
ij *» «/-. c'
and therefore
n") fH (x) H fa) = DXDX Rx> a , (N)
' i)"i. » C
and there are^> equations of this form, in which xlt ..., xp occur instead of xr.
If we determine ,%\', ... , x'p_l by the congruences
so that $1, ..., xfp,1 are the other zeros of a ^-polynomial vanishing in xlt ..., a?7,_1,
we can infer p — 1 other equations, of the form
i j *i- , a '
where r=l, 2, ...,(/»- 1). Here the right-hand side does not depend upon the place x.
And we can obtain p such sets of equations.
We have then sufficient * equations. For the hyperelliptic case the final formula is
given below (§ 217, Chap. XL).
198. Ex. i. Verify the formula (N) for the case p = 1.
Ex. ii. Prove that
is a rational function of x, xly ..., xp.
Ex. iii. Prove that if
then
Deduce the first formula of § 193 from the final formula of § 196.
* The function %\j(u), here employed, is remarked, for the hyperelliptic case, by Bolza,
Gottinger Nachrichten, 1894, p. 268.
198] ALGEBRAICAL THEORY OF THE THETA FUNCTIONS. 295
Ex. iv. Prove that if
ei=r£i.«i + ...... + rxc>;»'\
where a^ , . . . , at> are arbitrary places, and
V = Vx> m- VX}> mi— - Vx>"m>' = VCi' m - Vx'-1' ™l - ... - VXi>>" m>'
v f r ,. r ,. r
then
VJ— Wff • f v -r ^
017 — " Vct > c/r>-*'t, 1> •••> •*% p/>
where W denotes the function used in Ex. iv. § 125 ; it follows therefore by that example,
that „ ;;* = ^J£ . Hence the function
o Vr o YI
<^F1 + ...... + QpdVv
is a perfect differential ; it is in fact, by the final equation of § 196, practically equivalent
to the differential of the function log 6 ( Vx> m - VXl< m' - ...... - VXf' '""). Thus the theory
of the Riernann theta functions can be built up from the theory of algebraical integrals.
Of. Noether, Math. Annal. xxxvu. For the step to the expression of the function by the
theta series, see Clebsch arid Gordan, Abelsche Functionen (Leipzig, 1866), pp. 190 — 195.
Ex. v. Prove that if
(m2, Xi, j, ..., x\p, zlt ...,zp) = (a2, m*, .. , mj(2)
then
Ex. vi. Prove that
- 2 w(2)[^(^'m-^"m'- ...... -ux»m')-Ci(ua'm-v*l'mi- ...... -ux"'m^]
i = l
f-,x, a , , \
= FZ -^(x,a; z,xlt ..., xp).
Ex. vii. If
prove that
log ^ (ux'm- ux» m> - ...... -ux" m>)
= A+Alu*'a + ...... +ApUp'a+ I
where A, Alt ..., Ap are independent of x.
Ex. viii. Prove that
- 2 Mr (oO pf, r(«*'w -«*''"'>- ...... -M*>.»")= 2 tr
»• = ! r = 1
where a, c are arbitrary places and the notation is as in § 193.
[199
CHAPTER XL
THE HYPERELLIPTIC CASE OF RlEMANN's THETA FUNCTIONS.
199. WE have seen (Chap. V.) that the hyperelliptic case* is a special
one, characterised by the existence of a rational function of the second
order. In virtue of this circumstance we are able to associate the theory
with a simple algebraical relation, which we may take to be of the form
7/2 = 4 (as- fll) ... (x - ap) (x - c,}...(x- cp+l}.
We have seen moreover (Chap. X. § 185) that in the hyperelliptic case, when
p is greater than 2, there are always even theta functions which vanish
for zero values of the argument. We may expect, therefore, that the investi
gation of the relations connecting the Riernann theta functions with the
algebraical functions will be comparatively simple, and furnish interesting
suggestions for the general case. It is also the fact that the grouping of
the characteristics of the theta functions, upon which much of the ultimate
theory of these functions depends, has been built up directly from the
hyperelliptic case.
It must be understood that the present chapter is mainly intended to
illustrate the general theory. For fuller information the reader is referred to
the papers quoted in the chapter, and to the subsequent chapters of the
present volume.
* For the subject-matter of this chapter, beside the memoirs of Bosenhain, Gopel, and
Weierstrass, referred to in § 173, Chap. X., which deal with the hyperelliptic case, and general
memoirs on the theta functions, the reader may consult, Prym, Zur Theorie der Functionen
in einer zweiblattrigen Flache (Zurich, 1866) ; Prym, Neue Theorie der ultraellip. Funct.
(zweite Aus., Berlin, 1885); Schottky, Abriss einer Theorie der Abel. Functionen von drei
Variabeln (Leipzig, 1880), pp. 147 — 162 ; Neumann, Varies, iiber Riem. Theorie (Leipzig, 1884) ;
Thomae, Summlung von Formeln welche bei Amvendung der . . Roscnhaiii'schen Functionen gebraucht
u-erden (Halle, 1876) ; Brioschi, Ann. d. Mat. i. x. (1880), andt. xiv. (1886) ; Thomae, Crelle, LXXI.
(1870), p. 201 ; Krause, Die Transformation der hyperellip. Funct. erster Ordnung (Leipzig, 1886);
Forsyth, " Memoir on the theta functions," Phil. Trans., 1882 ; Forsyth, " On Abel's theorem,"
Phil. Trans., 1883 ; Cayley, "Memoir on the . . theta functions," Phil. Trans., 1880, and Crelle,
Bd. 83, 84, 85, 87, 88; Bolza, Gottinger Nachrichten 1894, p. 268. The addition equation is
considered in a dissertation by Hancock, Berlin, 1894 (Bernstein). For further references see the
later chapters of this volume which deal with theta functions.
200]
THE ZEROS.
297
200. Throughout this chapter we suppose the relative positions of the
branch places and period loops to be as in the annexed figure (4), the branch
place a being at infinity.
Fig. 4.
In the general case, in considering the zeros of the function S (ux> m — e),
we were led to associate with the place m, other p places mlt ..., mp, such
that *b(ux> m) has ml, .,., mp for its zeros (Chap. X. § 179). In this case we
shall always take m at the branch place a, that is at infinity. It can be
shewn that if 6, 6' denote any two of the branch places, the p integrals
Hi , ... , Up are the p simultaneous constituents of a half-period, so that
ur' = ni! &>r, i + + mpa>,.t p + m^w'r^ l + + WpVr, p, (r = 1 , 2, . . . , p),
wherein mly ..., nip, m/, ..., mp are integers, independent of r ; this fact we
shall often denote by putting w&'6' = £fl. It can further be shewn that if,
6 remaining any branch place, 6' is taken to be each of the other 2p + l branch
places in turn, the 2p + 1 half-periods, ub> v, thus obtained, consist of p odd
half-periods, and p + 1 even half-periods. Thus if the branch places, &', for
which ub'b' is an odd half-period be denoted by 61} ..., bp, we have, necessarily,
S- (?i&. &i) = 0, ... , S- (ub' V) = 0, and we may take, for the places m, mlt ..., mp,
the places b, b1} ..., bp. In particular it can be shewn that, when for 6 the
branch place a is taken, and the branch places are situated as in the figure
(4), each of ua> ai, ..., ua> ap is an odd half-period. We have therefore the
statement, which is here fundamental, the function <&(ux> a — ux*> a> — . . . — uxv ap)
has the places xl} ..., xp as its zeros. It is assumed that the function
S- (ux< a) does not vanish identically. This assumption luill be seen to be
justified.
For our present purpose it is sufficient to prove (i) that each of the
integrals ub> y is a half-period, (ii) that each of the integrals ua> a>, . . . , ua' "P is
an odd half-period. In regard to (i) the general statement is as follows: Let
the period loops of the Riemann surface be projected on to the plane upon
which the Riemann surface is constructed, forming such a network as that
represented in the figure (4) ; denote the projection of the loop (ar) by (Ar),
and that of (br) by (Br), and suppose (Ar), (Br) affected with arrow heads, as in
298
THE ZEROS.
[200
the figure, whereby to define the left-hand side, and the right-hand side ;
finally let a continuous curve be drawn on the plane of projection, starting
from the projection of the branch place b' and ending in the projection of the
branch place b ; then if this curve cross the loop (Ar) mr times from right to
left, so that mr is either + 1 or — 1, or 0, and cross the loop (Br) mr' times
from right to left, we have
b,b'
ur = wijft),^ j 4-
-f
mptor>p.
Thus, for instance, in accordance with this statement we should have
«r * — — »'r, u and uCr'ai = a>ri 1 — a)r> 2, and it will be sufficient to prove
the first of these results ; the general proof is exactly similar. Now we can
pass from cx to a1; on the Riemann surface, by a curve lying in the upper
Fig. 5.
sheet which goes first to a point P on the left-hand side of the loop (6j),
and thence, following a course coinciding roughly with the right-hand side of
the loop (ttj), goes to the point P', opposite to P on the right-hand side of
(61), and thence, from P', goes to au. Thus we have
=u —
On the other hand we can pass from d to ax by a path lying entirely in the
lower sheet, and consisting of two portions, from ca to P, and from P' to Oj,
lying just below the paths from cx to P and from P' to Oj, which are in
the upper sheet. Thus we have a result which we may write in the form
1( c, , P,
ur = (u
/ &i, -P\/
(ur ) .
f(x 1) _
But, in fact, as the integral ux> a is of the form I * — ?~l dx, and y has
different signs in the two sheets, we have
P,
,
(u
P,CI
, , a^P'., a-i, P"
} and (ur ) = — ur
201] THE HALF-PERIODS. 299
Therefore, by addition of the equations we have
u. = — a)
r, It
which proves the statement made.
In regard now to the proof that ua- °», . . . , ua> "v are all odd half-periods, we
clearly have, in accordance with the results just obtained,
tf ai = <oft i - (t»rt i+l + a>'r> i+1) - ...... - («,._ p + a,',, „) + (0,',, j + ...... + m'fi p),
which is equal to
(m'r, 1 + to'r, 2 + ...... + 0)'r>i) + (ft,,, i — ft)rj i+1 - ...... — ft),; p\
and if this be written in the form
Wjft),., j + ...... + mpco,.t p + mi'to'r, i + ...... -1- m'pto'r, p
we obviously have WjW/ + ...... + mpmp' = 1.
Ex. i. We have stated that if b be any branch place there are p other branch places
&!, 62, ..., bp, such that ub' \ ub< 6% ..., ub> b* are odd half-periods, and that, if b' be any
branch place other than 6, 6, , ... , 6P> w6' 6' is an even half- period. Verify this statement in
case p = 2, by calculating all the fifteen, =£6 . 5, integrals of the form w6-6',and prove that
when b is in turn taken at a, c, c1? c2, «1} a.2 the corresponding pairs bly b2 are respectively
Prove also that
Ex. ii. The reader will find it an advantage at this stage to calculate some of the
results of the second and fifth columns in the tables given below (§ 204).
201. Consider now the 2^ + 1 half-periods ub>a wherein b is any of
the branch places other than a. From these we can form ( ^ + j half-
V 2 /
periods, of the form ub- a + ub>> a, wherein b, bf are any two different branch
places, other than a, and ( ^ j half-periods of the form ub< a + uv' a + uv, «,
where b, b', b" are any three different branch places other than a, and so
on, and finally we can form tP~ J half-periods by adding any p of the
half-periods ub> a. The number
is equal to -1 +*[(* + 1)*+']^, or to 2»-l, and therefore equal to the
whole number of existent half-periods of which no two differ by a period, with
300 THE ASSOCIATION OF THE HALF-PERIODS [201
the exclusion of the identically zero half-period ; we may say that this number
is equal to the number of incongruent half-periods, omitting the identically
zero half-period.
And in fact the 22^ — 1 half-periods thus obtained are themselves incon
gruent. For otherwise we should have congruences of the form
' ft
wherein any integral ubK>a that occurs on both sides of the congruence may
be omitted. Since every one of these integrals is a half-period, and therefore
ub«> a = — ub«' a, we may put this congruence in the form
llb> ' a + M6" a + ...... + M&"1' a = 0,
and here, since we are only considering the half-periods formed by sums of
p, or less, different periods, m cannot be greater than 2j>>. Now this con
gruence is equivalent with the statement that there exists a rational function
having a for an w-fold pole and having b^ ..., bm for zeros of the first order
(Chap. VIII. § 158). Since a is at infinity, such a function can be expressed
in the form (Chap. V. § 56)
(as, l)r + y(as, !)„,
and the number of its zeros is the greater of the integers 2r, 2p + 1 + s. Thus
the function under consideration would necessarily be expressible in the
form (x, l)r. But such a function, if zero at a branch place, would be
zero to the second order. Thus no such function exists.
On the other hand the rational function y is zero to the first order at each
of the branch places alf ..., ap,clt ..., cp, c, and is infinite at a to the (2p+ l)th
order ; hence we have the congruence
ua"a+ ...... + uap> a + uc> -a+ ...... + UCP> a + uc> a = 0.
202. With the half-period of which one element is expressed by
, p,
we may associate the symbol
(KI , K% , . . . , kp \
KI , A?2 , . . . , Kp I
wherein kg, equal to 0 or 1, is the remainder when m, is divided by 2. The
sum of two or more such symbols is then to be formed by adding the 2p
elements separately, and replacing the sum by the remainder on division
202] WITH THE BRANCH PLACES. 301
by 2. Thus for instance, when p= 2, we should write ( I + (m ] = I _) .
If we call this symbol the characteristic-symbol, we have therefore proved,
in the previous article, that each of the 22p — 1 possible characteristic-symbols
other than that one which has all its elements zero can be obtained as the sum
of not more than p chosen from 2p + 1 fundamental characteristic-symbols,
these 2p + I fundamental characteristic-symbols having as their sum the symbol
of which all the elements are zero. In the method here adopted p of the
fundamental symbols are associated with odd half-periods (namely those given
by ua> °' , . . . , ua' ap), and the other p+ 1 with even half-periods. It is manifest
that this theorem for characteristic-symbols, though derived by consideration
of the hyperelliptic case, is true for all cases*. We may denote the funda
mental symbols which correspond to the odd half-periods by the numbers
1, 3, 5, ..., 2p — 1, and those which correspond to the even half-periods
by the numbers 0, 2, 4, 6, ..., 2p, reserving the number 2/> + 1 to represent
the symbol of which all the elements are zero. Then a symbol which is
formed by adding k of the fundamental symbols may be represented by
placing their representative numbers in sequence.
Thus for instance, for p = 2, Weierstrass has represented the symbols
H
\uj
H H H
oi/ ui/ vou vooy voo
respectively by the numbers
1 3024 5;
j- i 11 /10\ /00\ /10
and, accordingly, represented the symbol I J, which is equal to ( j + (
/5\
by the compound number 02. The ( = 10 combinations of the symbols
\AJ
1, 3, 0, 2, 4 in pairs, represent the 2-^ — 6 symbols other than those here
written. Further illustration is afforded by the table below (§ 204).
In case p = 3, there will be seven fundamental symbols which may be
represented by the numbers 0, 1, 2, 3, 4, 5, 6. All other symbols are
represented either by a combination of two of these, or by a combination of
three of them.
It may be mentioned that the fact that, for^ = 3, all the symbols are thus represen table
by seven fundamental symbols is in direct correlation with the fact that a plane quartic
is determined when seven proper double tangents are given.
* The theorem is attributed to Weierstrass (Stahl, Crelle, LXXXVIII. pp. 119, 120). A further
proof, and an extension of the theorem, are given in a subsequent chapter.
302 NOTATION FOR THE FUNCTIONS. [203
203. If in the half-period ^flm> m-, of which an element is given by
> p ^ i ...... tr, p,
we write \mg = Mit + ^ks, ^ms' = MS' + ^ks', where Ms, M,' denote integers,
and each of ks, ks' is either 0 or 1, we have (cf. the formulae § 190, Chap. X.)
where
X = [277 (M + W + 2r/ (M' + IF)] [it + to (M + P) -f to' (M' + P'
and therefore
* (u ; P, £ k') = e- *--«*• * (u + J Hm, ,„)•
The function represented by either side of this equation will sometimes be
represented by *t(u \^flm,^) ; or if inw,TO' =i*6" °-f *^'« + ...... +w&"«, the
function will sometimes be represented by ^ (u \ w&" a + ...... +ub*<a), or by
We have proved in the last chapter (§§ 184, 185) that every odd half-
period can be represented in the form
and, when there are no even theta functions which vanish for zero values of
the argument, that every even half-period can be represented in the form
in the hyperelliptic case every odd half-period can be represented in the
form
and every even half-period ^fT, for which ^(£fl') does not vanish, can be
represented in the form
and (§ 182, Chap. X.) the zeros of the function *t(ux'z |fl) consist of the
place z and the places n1} ..., np, while the zeros of the function ^ (ux> a \ ^O')
are the places blt ..., bp. In case p = 2 there are no even theta functions
vanishing for zero values of the argument ; in case p = 3 there is one such
function (§ 185, Chap. X.), and the corresponding even half-period £ fl" is
such that we can put
204]
TABLE.
303
wherein a?, is an arbitrary place and xz is the place conjugate to xl. Since
then ux<" °2 = — uXi> a*, this equation gives
now, as in 8 200. we easily find
O J
Ur' = — (tor, 3 + to'r, i + &>
and therefore
r, 2
r, a
Thus the even theta function which vanishes for zero values of the
argument is that associated with the characteristic symbol
In the same way for ^> = 4, the 10 even theta functions which vanish for
zero values of the argument are (§ 185, Chap. X.) associated with even half-
periods given by
where b is in turn each of the ten branch places.
204. The following table gives the results for p = 2. The reader is recommended
to verify the second and fifth columns. The set of p equations represented by the
equation (%Q.\ = ml«>r> l + m2<or, 2 + m/w',, i + «!•>',., 2 is denoted by putting fcQ = $ f™1 ^ \ .
\mlmzj
/. Six odd theta functions in the case p = Z.
Function
We have
Weierstrass's
number asso
ciated with
this symbol
Putting the corresponding half-
have for HI respectively
W>
*"* -*( $
02
(1)
«2
WiO
**-»'( Jl)
24
(3)
; • -
««,«»
*"<«-i(_*})
04
(13)
a
w>
— K-n)
1
(24)
c
9cei (u)
«c"c^(_!o)
13
(02)
c*
Sec, (U)
M -»(?.})
3
(04)
-
304
VERIFICATION OF THE THEORY
[204
77. Ten even theta functions in the case p = 2.
Function
We have
Weierstrass's
number asso
ciated with
this symbol
Putting the corresponding half-
period ^w6!' ai +u62> az, we
have for fcj , fc2
s«0
k( °°]
*( oo)
5
. «,,«, '
*.(*)
ua'c ~*\ oo)
23
(0)
VOO
1 / °°\
***-*( io)
12
(2)
C , C.J
W<0
1 /" 10\
~H oij
2
(4)
C , Cj
wo
I/ 10\
5v oo;
01
(12)
•,. «2,C1 „
fetf*
, /o o\
0
(14)
; «2)C2
V, 00
^'— *(-n)
14
(23)
«1)tfl
V*M
/ 01 \
~*\ ooy
4
(34)
*x
W«)
«c,0i =|^ Qij
34
(03)
«n «
•Sea, 00
1 ^-^
wc'ai=Hi o)
03
(01)
The numbers in brackets in the fourth column might be employed instead of the
Weicrstrass numbers ; they are based on the branch places according to the corre
spondence
1 302
But the Weierstrass notation is now so fully established that it will be employed here
whenever any such notation is used.
It should be noticed that the letter notation for an odd function consists always
of two a's or two c's ; the letter notation for an even function contains one a and one c.
The expression of the half-period associated with any function as a sum of not more
than two of the integrals ub- a, which has been described in § 202, is of course immediately
indicated by the letter notation employed for the functions.
Ex. Prove that if a =
.=«.<».
These equations effect a correspondence between five of the odd functions and the branch
places.
205]
OF THE LAST CHAPTER
305
205. Next we give the corresponding results for p = 3. Each half-period can be formed
as a sum of not more than 3 of the seven integrals ub- a (§ 202) ; the proper integrals
are indicated by the suffix letters employed to represent the function. We may also
associate the branch places with the numbers 0, 1, 2, 3, 4, 5, 6, say, in accordance with the
scheme
!> 3, 5, 0, 2, 4, 6;
then the functions ^ (w), &s(u),36(u) will be odd, and the functions 30(u), 92(u), 3t(u),S6(u)
will be even ; and every function will have a suffix formed of 1 or 2 or 3 of these numbers.
There is however another way in which the 64 characteristics can be associated with the
combinations of seven numbers, and one which has the advantage that all the seven
numbers and their 21 combinations of two are associated with odd functions, while all
the even functions except that in which the associated half-period is zero are associated
with their 35 combinations of three. It will be seen in a later chapter in how many ways
such a scheme is possible. One way is that in which the numbers
1, 2, 3, 4, 5, 6, 7
are associated respectively with the half-periods given by
, a + Uc3 , a
uc,
Uci>
By § 201 the sum of these integrals is = 0. The numbers thus obtained are given in the second
column. Further every odd half-period can be represented by a sum u^,a- u^ , «, - «w2, «2)
and all the even half- periods except one as a sum w^i, OI + M&S, «2 + M^3) «3 ; the positions of
nlt n2 or of b^ b2, 63 are given in the fourth column.
7. 28 odd theta functions for p = 3.
»•.(«)
^ttia.2 (U)
°ata3 (u)
1
2
3
12
13
23
74
75
001,
010
110
-
B,
20
306
(tt)
(«)
BY ACTUAL CALCULATION
Table I. (continued.}
[205
76
56
64
45
37
27
17
14
24
34
15
25
35
16
26
36
4
uc,a
/100
110
101
001
.
010
010
101
101\
10() 1
i
a 4. Ma, , a + w«2, « = f I 1
001
iioj
)
c ,
c , a3
c ,
C/7
3> W3
a , ^
a , c2
a , c3
a , c
205]
OF THE FUNCTIONS.
//. 36 even characteristics for p = 3.
307
bi bz b3
»(„)
, /000\
a, «2 a3
Ba^w
123
^•^.«+^.-iQ
*« r r
"J ^j *
MH)
456
uc>a =i(ooo)
ct c2 c3
3Cl (u)
567
Mc,,a =$(0°°}
C Ctf Co
**(«)
647
s*(oio)
C C3 Cj
KM
457
UCi'a sKooi)
1 2
W)
237
z«c, a + l<«i,« =1/^011\
2 \ 1 AA /
\xuwy
c a2 «3
v.<»)
317
^,a + Ma2)a ^|(°01)
C Q>q Oti
&*(•)
127
uo,a + ua3,a ^t™}
c a, «2
3clfll 00
234
=^ \oooj
**%*«)
314
»*H«- si(JJJ)
W«0
124
U0,a + Ua3,a ^iQ
Cl at a2
V, («)
235
Mc2) a + MBl, a ^(°00)
c, o, «3
Q /,,\
*!%•, V'4,'
315
W%,.+^,a =i(^)
o.2 03 fll
Sen (u)
125
Mfi.,a + M«,,« -i^01^
5 \oiiy
c2 «! a2
*w, (»)
236
^..^- si(JJJ)
i*M
316
««„« + ?<«, a -if00°^
c3 a3 «j
»W)
126
=^ \oooy
3 1 ^2
^«lC2C3 (M)
156
76a,,a + w(;2,a + ^3>« =i(J}J)
ft f* f*
1 1
SM^W
164
^.a + ^a + ^^Q
Cti Co C
5ftlClC,(«)
145
*..+*,.+*.. .jQ
5-'(M)
147
Ma1,a + 7/c,a + „<-, , a = ^^ j
20—2
308
METHODS OF NOTATION.
[205
Table II. (continued).
V»H
157
ft_ /111\
= * \110J
•9a,cc3 («)
167
*••+*• +«fc.«iQ
•satc.£3 (U)
256
-^-^--iQ
«2 ^ C
*W>(«0
264
^,.+^..+^.•4^
•J(l2C]C.2 (^)
245
a 4- c rt-4^°10\
~^ yiooy
«2 C3 C
5a (w)
247
2^(12 ) ^ -^- ^C, (I -1- 1(C\ , (1 = i I j
S«.2cc.2 («)
257
•^<l-2 , d -|- ^C, rt -i- 1(,C'2 j & = -4 I
V^M
267
7^CE-2 » ^ -|- ?£C, Ct -J- ^^3 , (I = i [ J
AM.M
356
""•'••H"»*<H*'««*(J?J)
a3 Cj c
*W. 0)
364
^•+—+-..s *Q
rto <^2 "
VMM
345
-'^-^— Km')
v_ei
347
^.^^.a+^.a^lQ
Ct 0 CtJ ^?0
*M«<>)
357
•%.•+«• VM.«-I^)
**(.)
367
«...+«. +—iQ
It is to be noticed that every odd theta function is associated with either (i) any
single one of at, a2, «3 or (ii) any pair of ax, a2, a3 or any pair of c, Cj, c2, c3, or (iii) a
triplet consisting of one of c, cu c2, c3 and two of an a2, a3 or consisting of three from
c, c1? c2, c3. This may be stated by saying that odd suffixes are of one of the forms
or, a2, c2, a2c, c3. Similarly an even suffix is of one of the forms c, ac, ac2, a3.
In the tables just given the fundamental characteristic-symbols, denoted by the num
bers 1 , 2, 3, 4, 5, 6, 7, are those associated with sums of integrals which may be denoted by
206] EXPRESSION OF THETA QUOTIENTS. 309
We can equally well choose seven fundamental odd characteristic-symbols, associated with
the integrals denoted by any one of the following sets :
c Cj , c c2 , c c3 , c «2«3, c aza^ c «1a2> Ciczc3
e2c , CgCj, c2c3,
The general theorem is — it is possible, corresponding to every even characteristic « , to
determine, in 8 ways, 7 odd characteristics a, /3, y, *, A, /z, v, such that the combinations
a, ft y, *, A, M, «*, f«j3, ea/c, eX/x
constitute all the 28 odd characteristics, and the combinations
f, a/3y, a*X, /3y*c
constitute all the 36 even characteristics. In the cases above f =0. The proof is given in
a subsequent chapter.
206. Consider now what are the zeros of the functions
^ (w), & («| lA • « + ...... + M6*' a),
where blt ... ,bk denote any k of the branch places other than a (k $-p), and u
is given by
the functions being regarded as functions of xl.
The zeros of S- (u) are the places zlt ..., zp determined by the congruence
or, by
Provided the places a,x2, ..., xp be not the zeros of a ^-polynomial, that is,
provided none of the places x2> ...,xp be at a, and there be no coincidence
expressible in the form #» = #;, the places zlf z2, ...,zp cannot be coresidual
with anyp other places (Chap. VI. § 98, and Chap. III.) and therefore (Chap.
VIII. § 158) this congruence can only be satisfied when the places z1} ...,zp
are the places
Cl, X% , X3 , . . . , Xp ,
these are then the zeros of ^ (u), regarded as a function of x^
The two places for which .r has the same value, and y has the same value with opposite
signs, are frequently denoted by x and x.
310 EXPRESSION OF THETA QUOTIENTS [206
The zeros of *b(u\ub>'a + ...... + ubk>a) are to be determined by the
congruence
Ux"ai + ...... +Uxi»a" + Ubl> a + ...... + Ubk'a = Ux" a — Mz" a> — ...... — Uzi» ai>,
or, by
w*i, &i + W*2, x* + ...... + u*p, *i> + M6«, « 4. ...... 4. M6fc, a = 0,
which we may write also
(z1} z,, ..., zp, a*"1) = (blt . . . , bk, x.2) . . . , xp) ;
in particular the zeros of S- (u ub> a) are the places b, x.2, ...,xp.
207. Now, in fact, if the sum of the characteristics qly ..., qn differs from
the sum of the characteristics r1} . .., rn by a characteristic consisting wholly
of integers, n being an integer not less than 2, then the quotient
; rO ...... *(*; rn)
is a periodic function of w.
For, by the formula (§ 190, Chap. X.)
^ (w + n,n; g) = ex»(w) + 27rz'()ft9' - m'«> ^ (it ; q),
where m denotes a row of integers, we have
f(u + flm) _ 2iri [wi(2(?, _ 2r/) _ m, (2 _ 2r)-j
755
and if S^' — Sr', Sg — 2r, each consist of a row of integers the right-hand
side is equal to 1.
Hence, when the arguments, u, are as in § 206, the function f(u) is a
rational function of the places xlt ..., xp.
208. It follows therefore that the function
*•(*)
is a rational function of the places xl, ...,xp. By what has been proved
in regard to the zeros of the numerator and denominator it has, as a function
of a?!, the zero b, of the second order, and is infinite at a, that is, at infinity,
also to the second order. Thus it is equal to M (b — x^, where M does not
depend on xv As the function is symmetrical in xlt x.2, ...,xp, it must
therefore be equal to K (b — x^) ... (b — xp), where K is an absolute constant.
Therefore the function
may be interpreted as a single valued function of the places xl} ...,xp,
on the Riemann surface, dissected by the *2p period loops. The values of
the function on the two sides of any period loop have a quotient which is
constant along that loop, and equal to + 1.
209] BY MEANS OF RATIONAL FUNCTIONS. 311
The function has been considered by Rosenhain*, Weierstrass f, RiemannJ and
Brioschi§. We shall denote the quotient 3 (u\u!>- *} I S (u} by qb (u). There are 2p + l such
functions, according to the position of b. Of these ^ (u), ..., qaj> (u) are odd functions,
and q (u), qc (u}, ..., qc (u) are even functions. The functions are clearly generalisations
of the functions \/x = sn u, \A~ # = cn u, \/l — fctx = dn u, obtained from the consideration
of the integral
dx
209. Consider next the function
b» a
* (111 1 ->ib, , a\ ^V ( ii i -i/fyfcj a\
r \ *»I«F»* / »j \ M" , "< /
wherein 61; ..., 6jfc are any A; branch places other than a. We consider only
the cases k < p + 1. By what has been shewn, the function is rational in xlt
and if zlt ..., zp denote the zeros of ^ (u ub»a+ +&*.«) the zeros of the
numerator, as here written, consist of the places
j_j k—\
y IV fjK 1 nn /y
-- I , . . • , ^ i/ 1 U , ' J , • • • > />
and the zeros of the denominator consist of the places
Thus the rational function of xl has for zeros the places zlt ..., zp, ak~l,
and, for poles, the places blt ..., &&, x.2, ..., xp. It has already been otherwise
shewn that these two sets of p + k - 1 places are coresidual. Now any
rational function, of the place x, which has these poles, can (Chap. VI. § 89)
be written in the form
uy + v (x - 6Q . . . (x - bk)
(x — &i) . . . (x — bk) (x — x.^...(x — xp) '
wherein u, v are suitable integral polynomials in x, so chosen that the
numerator vanishes at the places x2, ..., xp. The denominator, as here
written, vanishes to the second order at each of blt ..., bk, and also vanishes
at the places x2, x2, ..., xp, xp.
Let X, //, be the highest powers of x respectively in u and v. Then, in
order that this function may be zero at the place a, that is, at infinity, to the
order &— 1, it is necessary that the greater of the two numbers
* Memoires par divers savants, t. xi. (1851), pp. 361 — 468.
t By Weierstrass the function is multiplied by a certain constant factor and denoted by al(u).
£ In the general form enunciated, as a quotient of products of theta functions, Werke
(Leipzig, 1876), p. 134 (§ 27).
§ Annali di Mat. t. x. (1880), t. xiv. (1886).
312 DEDUCTION OF RELATIONS [209
(wherein 2 ( p + k - 1) is the order of infinity, at infinity, of the denominator)
should be equal to — (& — !). Since one of these numbers is odd and the
other even, they cannot be both equal to —(&-!). Further in order that
the ratios of the A, + /* + 2 coefficients in u, v may be capable of being chosen
so that the numerator vanishes in the places x.2, ..., xp, it is necessary that
X + M + 1 should not be less than p— 1. And, since a rational function
is entirely determined when its poles and all but p of its zeros are given,
these conditions should entirely determine the function.
In fact we easily find from these conditions that the case 2\+2p+ 1 >2(/i-f&)
can only occur when k is even, and then X = %k — 1, /M =p — 1 — ^k, and
that the case 2X + 2p + 1 < 2yu, + 2k can only occur when k is odd, and then
X = £ (k — 3), p = p — ^ (k + 1). In both cases X + /* + 2 =p.
By introducing the condition that the polynomial uy + v(x-b1) ... (a - bk)
should vanish in the places #2, ..., xp we are able, save for a factor not
depending on #, y, to express this polynomial as the product of (x— ^^...(x— &*)
by a determinant of p rows and columns of which, for r > 1, the rth row is
formed with the elements
A-l
r yr yr
wherein $(x) denotes (x — b^ . . . (x — bk), the first row being of the same
form with the omission of the suffixes.
Therefore, noticing that F is symmetrical in the places asl} ..., a;p, we
infer, denoting the product of the differences of ar1} ..., xp by A(#i, ..., xp),
that
lavy, £jfr Vr M M-I J
*-1> ""''
where (7 is an absolute constant, and the numerator denotes a determinant
in which the first, second, ... rows contain, respectively, xlt x2> ...; and here
when & is even, \ = ^k — 1, p=p — 1 — 1&
and when k is odd, X = ^(& — 3), /* = jj — ^ (k + 1).
210. By means of the algebraic expression which we have already
obtained for the quotients ^ (u ub< a)/^ (u), we are now able to deduce an
algebraic expression for the quotients
(u) ;
since it has already been shewn that by taking k in turn equal to 1, 2, ...,p,
and taking all possible sets b1} ..., bk corresponding to any value of k, the
half- periods represented by ubi>a+ ...... -\-ub*>a consist of all possible half-
periods except that one which is identically zero, it follows that, in the
210] CONNECTING THE THETA FUNCTIONS. 313
hyperelliptic case, if u denote ux><a> + ...... + uxi»a», and q denote in turn all
possible half-integer characteristics except the identically zero characteristic,
all the 2^-1 ratios ^ (u ; q)^ (u) can be expressed algebraically in terms of
xl} ..., xp, by the formulae which have been given.
The simplest case is when k = 2 ; then we have A, = 0, //, = p - 2, and
_
: :
_
- bj (xr - 6,) jRT^) '
where ^ (a?) = (x - x^ (x - #2) . . . (x - xp}, and C is an absolute constant.
Denoting the quotient ^ (u ub»a + ifr, •)$ (u) by qbii &2, we have
=i
-— -
- &i) (a?r - 62) .R' («r)
where ^Ilj2 is an absolute constant; and there are p(2p + l) such
functions.
When k = 3, we have X = 0, n =p - 2, and, if qbl>bl>bt denote the quotient
£ (u\ub»a + ifa a + ub» a)/^ (u), we obtain
^> v 1
= — — ---
where BJt 2i 3 is an absolute constant. It is however clear that
<?&! , 62 <?&i,&3 _ /; 7 x g&, ,62,63
so that the functions with three suffixes are immediately expressible by those
with one and those with two suffixes.
More generally, the 2* - 1 quotients %(u; q)/$ (u), depending only on
the p places xlt ...,xp, must be connected by 22P-p-l algebraical rela
tions; and since (Chap. IX.) any argument can be expressed in the form
was"°' + ...... +M*P.«P, it follows that these may be regarded as relations
connecting Riemann theta functions of arbitrary argument. This statement
is true whether the surface be hyperelliptic or not.
Of such relations one simple and obvious one for the hyperelliptic case under con
sideration may be mentioned at once. We clearly have
and therefore
^^5M3 (u) 3bl (w) + ^-l^3&1 (u) \(u) + b-^* 3blbtW3bt(u),= 0.
It is proved below (§ 213) that A'^ : A\ : A'\2 = (b.,~b3) : (b3~bj : (b^bj.
Other relations will be given for the casesjo = 2, p = 3. A set of relations connecting
the y's of single and double suffixes, for any value of p, is given by Weierstrass (Crelle LII
Werke I. p. 336).
314 CASE WHEN THE ARGUMENT IS [211
211. Ex. i. Prove that the rational function having the places xlt ..., xp, a, as poles,
and the branch place b as one zero, is given by
where R (£) = (£ — x} ({- — x^ (^-^p)> and, in the summation, x0, y0 are to be replaced
by x, y.
Prove that if u denote the argument
then
52 (M u<>- «) Z2
— ^.> / v — — A
(b~x)(b-x1) ...... (b-xvy
where A is an absolute constant.
Prove for example, in the elliptic case, with Weierstrass's notation, that
Ex. ii. If Zr denote the function Z when the branch place br is put in place of 6, and
R (br) denote (br — x) (br- Xj) ...... (br — xp\ and we put
=
prove that
-r A (A', ^j, ..., ^p),
where B is an absolute constant, A (A; xlt ..., xp) denotes the product of all the differences
of the (p + l) quantities x, x^ ..., xp, <j) (xr) = (xr — b^) ...... (xr—bk), and the determinant is
one of p + l rows and columns in which, in the first row, x0, y0 are to be replaced by x, y.
Prove that, when k is even, X = J(£-2). /*= P~\k, and, when k is odd, X = |(£ — 1),
Ex. iii. Hence prove that the function
multiple of
A(.r, j?ls ..., J7P)
This formula is true when k = \.
Ex. iv. A particular case is when £ = 2. Then the function 5 (M ubl>a + ub2' a)/3(u) is
a constant multiple of
wherein /2 (
Ex. v. Verify that the formula of Ex. iii. includes the formulae of the text (§ 210) ;
shew that when x is put at infinity the values of X, /x in the determinant of § 209 are
properly obtained.
211] A SUM OF p + l INTEGRALS. 315
Ex. vi. Verify that the expression ^(x^b; a,x1, ..., xp) of § 130, Chap. VII., takes
the form given for the function Z of Ex. i. when a is the place infinity.
Ex. vii. If f(x~) denote the polynomial
prove that any rational integral polynomial, F(x, z), which is symmetric in the two
variables x, z and of order p + l in each of them, and satisfies the conditions
is of the form
F(x,z)=f(x,z) + (x-z)*+(x,z),
where (cf. p. 195), with A0 = A, A2,) + 3 = 0,
P+I
f (x, z)=-S x^zi (2A2i + X2i + 1 (x+z)},
t=0
and ^ (x, z) is an integral polynomial, symmetric in x, z, of order/)- 1 in each*.
In case jo = 2, and f(x) = (x-a1) (x — a^)(x — c) (x-c1)(x — c2), prove that a form of
F(x, z) is given by
F (x, z) = (x- ctj) (x - 02) (z -c)(z- Cj) (z - c2) + (z - at) (z - «2) (x - c) (x -c^(x- c2).
Ex. viii. If for purposes of operation we introduce homogeneous variables and write
prove that a form of F(x, z) is given by
\P / 3
A*
where, after differentiation, jclt x2, zlt z2 are to be replaced by x, 1, z, 1 respectively.
This is the same as that which in the ordinary symbolical notation for binary forms is
J 1 J 1- J/ \ n P+1 P+l j-/ \ 1 • 2^+2
denoted by/ (x, z) = 2ax az , f(x) being ax .
Ex. ix. Using the form of Ex. viii. for F(x, z), prove that if elt e2, x, xly ..., xp
be any values of x, we have
s f^ i ^s f(x" x*> /(ei) /C^) 7(gi> %)
r=0 [0' S3F"1" ™ G" (o?r) 6r" (j?.) [(?' (6l)]2 T [>" (e2)]2 * »' (6l) G' (ej '
where G'(^) = (^-e1) (£ - e.2) (g - x) (i- — xj ...... (^ — xp),and the double summation on the
left refers to every one of the \p (p + l) pairs of quantities chosen from x, xlt ..., xp.
Ex. x. Hence it follows!, when yz=f(x\ yr2=/(-*V)> etc., and R(£) = (£-$) (^-^1)...
(I - XP\ that
R(^Rif^*- Jfe_ - T - /
W ^ ^ L 0 («i - '^r) (*2 - «r) ^ (ir) J '(«! - e
is equal to
* It follows that the hyperelliptic canonical integral of the third kind obtained on page 195
can be changed into the most general canonical integral, -R*1 a (p. 194), in which the matrix a
has any value, by taking, instead of/(x, z), a suitable polynomial F(x, z) satisfying the conditions
of Ex. vii.
t The result of this Example is given by Bolza, Gotting. Nachrichten, 1894, p. 268.
316 A SOLUTION OF THE INVERSION PROBLEM [211
where the summation refers to every pair from the p + l quantities x, x±, ..., zp, and
/(#, 2) denotes the special value of F(x, 2) obtained in Ex. viii.
Ex. xi. It follows therefore by Ex. iv. that when bt , b.2 are any branch places of the
surface associated with the equation y*-f(%) = 0, there exists an equation of the form
V (u\ub»a + ub»a)_ 2yryg-f(xr,xg) f (blt b,}
&W~ (l) ' G'(xr}G'(xg} ~(V-~^?'
where C is an absolute constant, (*(£) = (£ — bj (£ - b2) (g — a;) (g - x^) ...... (%-Xp), and
u = ux> a + uXl'cll + ...... +UXP' aP. The importance of this result will appear below.
212. The formulae of §§ 208, 210 furnish a solution of the inversion
problem expressed by the p equations
*!,<*! Xp,dp / * -I C» \
Ui + ...... +Ui =ut; (»— 1, 2, ..., j>>
For instance the solution is given by the 2p + 1 equations
from any p of these equations xl} ...,xp can be expressed as single valued
functions of the arbitrary arguments ult ..., up.
And it is easy to determine the value of A2. For let blt ..., bp, 6/, ..., bp'
denote the finite branch places other than 6. As already remarked (§ 201)
we have
\c, GI, . . . , Cp) = (ft, a/1, . . . , Up)
and therefore
(b, b1} ...,bp) = (ft, bi, ..., bp').
Now we easily find by the formulae of § 190, Chap. X. that if P be a set
of 2p integers, Plt ..., Pp, P/, ..., P/,
^(u)
n.pp,
hence, if ub' a = % ^ v , and u0 = w6' - a + ...... + ubp< a, we have, by the formula
under consideration, writing b1} ..., bp in place of xl} ..., xp, the equation
and, writing &/, ..., bp in place of xlt ..., xp, we have
^•2 /,/ i n.b, a »,b, a\
r* \uo ^ It ' \lt ' ) . n , ,. n
^(^ + ^'g) /i(6"6l)-'(6
thus, by multiplication
e-«pi> = A* (& _ &i) . . . (6._ bp) (b _ 6;) ..
213] IN THE HYPERELLIPTIC CASE. 317
and hence
W(u\ub> a) (b-xl)(b-x^...(b-xp)
where/(#) denotes (x — a^) ... (x — ap) (x — c}(x — cl}...(x — cp), and eirit>p = ± 1
according as ub>a is an odd or even half-period.
The reader should deduce this result from the equation (§ 171, Chap. IX.)
U • t •*, \ V(TT TT • £ -., \ — ^ ~~^(xi)} ...... (^ ~^(xv)}
.... UP, &,yi) ...... r(»V-i V* **>y*>-
by taking Z to be the rational function of the second order, x.
'a> + ...... + uxv> aP, we deduce (see Ex. i. § 211)
yr 1
If in particular we put b in turn at the places a1} ..., ap, write
P (x) = (x-al) ... (x - ap) and Q (x} = (x - c) (x - cx) . . . (x - cp), and use the
equation
(x - a?x) . . . (x - xp} __ | (at - x^ . . . (af - Xp)
~P(x) hf~ (a>-di)P(at)
we can infer that oc1} ..., xp are the roots of the equation*
t=i
where e^ is + 1 and is such that we have
<&_(U tt«. a) (at - Xj _ (dj - Xp}
Another form of this equation for xlt ..., xp is given below (§ 216), where
the equation determining yt from x± is also given.
213. We can also obtain the constant factor in the algebraic expression of the
(u\ubl'a+ub*'a)3(u) + 3 (u ?«6"a) 3 (u\u^' a}.
Let />j , b.i denote any branch places, and choose zlt ..., zp so that
then zlt ..., zpta are the zeros of a rational function which vanishes in ^, ..., .rp, 6,.
Such a function can be expressed in the form
* Cf. Weierstrass, Math. Werke (Berlin, 1894), vol. i. p. 328,
318 DETERMINATION OF A CONSTANT FACTOR. [213
where (x, l^7"1 is an integral polynomial in x whose coefficients are to be chosen to satisfy
the p equations
-yi + (#i-6i)fo, IX'-^O, (i = l, 2, ...,p) ;
thus the function is
where F(.r) = (x — x1) ... (x-x^) ; and, if the coefficient of x2P + 1 in the equation associated
with the Riemann surface be taken to be 4, we have
^-<*-^t*(4p[£^
and therefore, putting bt for x,
y*- i T
' '
Now we have found, denoting ux>! a> + ...... + u*** ' °P by u, and uZl> a' + ...... + w2" ' a" by v,
the results
where w^2' a = ^Qp p-; hence we have
1* y* JLT
which, by the formulae of § 190, is the same as
*» - y<
_
5 (u\ub>< a) 5 (M|M6«- a)
where e is a certain fourth root of unity.
Thus the method of this § not only reproduces the result of § 210, but determines the
constant factor.
Ex. Determine the constant factors in the formulae of §§ 208, 210, 211.
214. Beside such formulae as those so far developed, which express
products of theta functions algebraically, there are formulae which express
differential coefficients of theta functions algebraically; as the second
differential coefficients of ^(w) in regard to the arguments u^, ..., up are
periodic functions of these arguments, this was to be expected.
We have (§ 193, Chap. X.) obtained* the formula
-«, I
= T .
J-*i i — . ~K » i_\~K9 "v \™K> A~/J J^
*=i at
Cf, also Thomae, Crelle, LXXI., xciv.
214] EXPRESSION OF THE £ FUNCTION. 319
we denote by hr the sum of the homogeneous products of xl , ..., xp, r together,
without repetitions, and use the abbreviation
further, for the p fundamental integrals MX ' M, . . . , up' M, we take the integrals
f * dx xx dx x xP'1 dx
dx fxx dx fx xP'1 dx
y ' Jp y ' ' *'/p y
then it is immediately verified that
i&k', xl} ...,Xp)/dxk
~ '"*
where jP (a;) denotes (# — a,-,) ... (x — xp).
Thus, if p, v denote the values of x and y at the place /z, we have, writing
a, Oj, ..., 03, for w, Wj, ..., w^ (§ 200),
- & (M*. ° - ux>< a' - ...... - W*P. a^) + §;-(t^. « - «*..«. - ...... -UXP< ap}
T*>» , i gxp-'-fo*; ^i* •••>ay) fy + y* y* + H .
*•%
therefore, also, the function
is equal to
.. .. _
t=i JT 0*) p, - xk
which is independent of the place x.
Now let R (t) denote (t — x)(t—x1) ... (t—xp), and use the abbreviation
given by the equation
, - , ...,
'
then also
, , -- , .- _ , ,
1^ (xp) Jp-{~1 ^ l ' • • • '
Now
is equal to
(«, A-'i 4- A-2) -
320 EXPRESSION OF THE f FUNCTION. [214
wherein kr denotes the sum of the homogeneous products of x2, ..., xp,
without repetitions, r together, and is therefore equal to
or to
(Xi X) J£p_i_i {Xi ', X2> '•• > xy>}'
Hence
(x, - x) Ff (x,)
_
F'(x,) x-Xi F' '(a?,)
While, also,
Thus
^_t , !,..., y - - „, , .
k=l * \-'k/
Therefore the expression
is equal to
In this equation the left-hand side is symmetrical in x, x1} ..., xp, and the
right-hand side does not contain x. Hence the left-hand side is a constant
in regard to x, and, therefore, also in regard to x1} ..., xp. That is, the left-
hand side is an absolute constant, depending on the place /*. Denoting this
constant by — C we have
' a
i,--> Xp-l , „
_ _ _
2R' (x) 2R' (xp)
215. From this equation another important result can be deduced. It
is clear that the function
does not become infinite when x approaches the place a, that is, the place
infinity. If we express the value of this function by the equation just
obtained, it is immediately seen that the limit of
215] BY ALGEBRAICAL INTEGRALS AND RATIONAL FUNCTIONS. 321
and that the expression
when expanded in powers oft by the substitutions ac = -,y = —^ (1 + At2 + ...),
"
t
where A is a certain constant, contains only odd powers of t. Hence the
limit when t is zero of the terms of the expansion of this expression other
than those containing negative powers of t, is absolute zero, and therefore,
does not depend on the places xl} ...,xp. The terms of the expansion which
contain negative powers of t are cancelled by terms arising from the integral
L^ . Since this integral does not contain xl} ..., xp we infer that the
difference
rx * _ yxp-j(®', BI, --, a?)
2R' (x)
has a limit independent of xl} ..., xp, and, therefore, that
no additive constant being necessary because, as & (u) is an odd function,
both sides of the equation vanish when xlt ..., xp are respectively at the
places Oj, ...,ap. As any argument can be written, save for periods, in the
form ux» a> + . . . +UXP> av, this equation is theoretically sufficient to enable us to
express £ (w) for any value of u.
Ex. i. It can easily be shewn (§ 200) that
uc'a + uc>'a< + ...... -\-uc*»aP = Q.
Thus the final- formula of § 214 immediately gives
...... ......
fc=i 2 (xjc- c) F' (xk)
Ex. ii. In case p = \ we infer from the formula just obtained, and from the final
formula of § 214, respectively, the results
where D is an absolute constant. Thus
X 96 1
This is practically equivalent with the well-known formula
The identification can be made complete by means of the facts (i) The Weierstrass
argument u is equal to ua> x, in our notation, so that y=-$>' (u), (ii) ux< a' = (a + o>'-u, so
that ^(ux- "0 = ^ (« + «'-«)= -4'"'=- r*£^? , as we easily find when Z*'M is
7 rti y
B. 21
322 EXAMPLES. [215
chosen as in § 138, Ex. i., (iii) d£u = —— , (iv) therefore fx (ux' a>) = - £u, (v) the branch
i7
places clt alt c are chosen by Weierstrass (in accordance with the formula ^+62 + 63=0)
so that the limit of $>u — ^, when u = 0, is 0. The effect of this is that the constant
D is zero.
Ex. iii. For p = 2 we have
- f, (ux> a+uXl' ai + ux*> a*) = Lx' » + L*» ^+Z*" *
-2 i--g *t--i c
A- - #2) 2 (#! — x} fa — #2) 2 (#2 - x) (xz - x^) l
and
-t *~ * o * «
where with a suitable determination of the matrix a which occurs in the definition of the
integrals Lx' M and in the function 3 (u), we may take (§ 138, Ex. i. Chap. VII.)
For any values of p we obtain
-fP0«*i>0l + ...... + uxP<ai>)=Lx»a> + ...... + Za!P>aP=-^±1I fx
4 *=ijak y
Ex. iv. We have (§ 210) obtained 22" - 1 formulae of the form
$(u\ub"a + ...... +ubk>a)
~~
where Z is an algebraical function, and the arguments ult ..., ut) are given by
the integrals being taken as in § 214, these equations lead to
Hence we have
For instance, when £ = 1, and Z is a constant multiple of ^(b^—x-^ ...... (bl — xp\ we
obtain
so that
a\_/*i, «i i -L/^I^P-V &r (r • r r}
/"•*'| + ...... + ,- * aJp/vT Jfi»-*-iw« *i' '"»*i»^
,.=1^^ ^r; [_
xr-b
_TXl,al , .TXp.Op | ^ Xp-i^r? ^, #1, ...,^p)
~ + ...... + --- "
216] ANOTHER SOLUTION OF THE INVERSION PROBLEM. 323
By means of the formula
which is easily obtained from the formulae of § 190, we can infer that the formula just
obtained is in accordance with the final formula of § 214.
(2»-f-l\
/
even theta functions which do not vanish ; and the corresponding half-periods are con
gruent to expressions of the form
It may be shewn in fact that these half-periods are obtained by taking for xv , . . . , xv the
( P ] possible sets of p branch places that can be chosen from o15 ..., ap, c, c1? ..., cp.
V P )
Hence it follows from the formula of the text (p. 321) that if $Qk be any even half-period
corresponding to a non- vanishing theta function, we have
This formula generalises the well-known elliptic function formula expressed by fw = ^.
To explain the notation a particular case may be given ; we have
£,(«,,,., <*2,r, ..-, *P,r)=r,i,r, ™ &<1*t*f)= -Z****'
and
fi(»',,r,«'2,r, »., »'ftr)=«?'<,r,or W* "*) = - L*' «*.
Thus each of the 2p2 quantities ^ ,., 7?'f, r can be expressed as ^-functions of half-
periods.
Ex. vi. The formula of the text (p. 321) is equivalent to
where
For example when p =
216. It is easy to prove, as remarked in Ex. iii. § 215, that if
and the matrix a (§ 138, Chap. VII.) be determined so that the integrals
L*'* have the value found in § 138, Ex. i., then
| [*ta?dx
i — - .
=lJ a y
a
Therefore, if - — %r (u) be denoted by $>r, i (u), we have
21—2
324 SOLUTION OF THE INVERSION PROBLEM [216
and thus, as follows from the definition of the arguments u,
where F (x) denotes (x — x^) ... (x — xp).
Whence, if a; be any argument whatever,
XP | xi_,
T? j i / \ i-x <c t = l
^ y Pi i \^)> — 4 ^2p+l — ~
i=l k=l
but we have
P p ._.
"
Thus
p 2 i-l , .
V/c = Z Xk K>PI i (u).
Thus, if we suppose Xap+i = 4, the values of a?1} ..., a?p satisfying the
inversion problem expressed by the equations
M = w*" a' -f ...... + w^' °v
are the roots of the equation
F (x) = XP- XV-* ppf p (M) - a^9fi ^(u)- ...... -p^ (u) = 0.
In other words, if the sum of the homogeneous products of r dimensions,
without repetitions, of the quantities xl} ...,xp be denoted by hr, we have
hr = (-y-1$>p>p-r+i(u).
Further, from the equation
-i fa', ^i ..... XP)
fat F'(xk)
putting p for i, we infer that
, - .
dup \_ ditp }x=Xk
because F(xk} = 0. Thus, if we use the abbreviation
we obtain
216] BY THE ip- FUNCTIONS. 325
These equations constitute a complete solution of the inversion problem.
In the ^-functions the matrix a is as in § 138, Ex. i., and the integrals of the
first kind are as in § 214.
We have previously (§ 212) shewn that xl} ..., xp are determinable from
p such equations as
^ (u | M«< >a)=v (at - ap . . . (en- xp) _(ai-xl)...(ai-xp)
V(u) ^-p'-^Qla,) K Say"
Thus we have p equations of the form
Ex. i. For p = 1 we have
This is equivalent to the equation which is commonly written in the form
• 3 9 / / \ *
sn2 (u v e1 — es)
v. ii. For p = 2 we have
We may denote the left-hand sides of these equations respectively by /i^2, Mq£.
Ex. iii. Prove that, with AflS^~<%IV»i(«)-ft.'t<4«ll^ MI= ± V^/K), we have
] + «1«2 [^22 («) ~ ^22 00]-
-. iv. Prove that
^. v. If, with P (^) to denote (x-a^ ...... (x - «p), we put
F = /"*• P(a?) 0^' r«p P (a:) d
Jaix-ar2y ]at>x-ar2
prove that
3 a _0 3
o'y/ T ...... ~r oT?" — * 5 — •
o ^ i 9 Vp dup
Ex. vi. With the same notation, shew that if
then
326 EXAMPLES. [216
The arguments Vl, ..., Vp are those used by Weierstrass (Math. Werke, Bd. i. Berlin,
1894, p. 297). The result of Ex. iv. is necessary to compare his results with those here
obtained. The equation yr = \js(xr) is given by Weierstrass. The relation of Ex. vi.
is given by Hancock (Eine Form des Additionstheorem u. s. w. Diss. Berlin, 1894,
Bernstein).
With these arguments we have
Ex. vii. Prove from the formula
-(i(ux
where
that the function
F(x)
is independent of the place x. Here c is an arbitrary place and F (x) = (x - x-^) ...... (x - x^).
Ex. viii. If ^ac denote the integral if ' * - 222^, ,• uz: c uf a, obtained in § 138, and
F^ a denote -D^' ", prove that in the hyperelliptic case, with the matrix a determined as
in Ex. i. § 138, when the place a is at infinity,
2 ;/* y
Hence, when A2P+1 = 4, shew that the equation obtained in § 215 (p. 321) is deducible
from the equation (Chap. X. § 196)
Ex. ix. We can also express the function £p(u+v)- £,,(11) - £p(v), which is clearly a
periodic function of the arguments u, v, in an algebraical form, and in a way which
generalizes the formula of Jacobi's elliptic functions given by
) = k* sn wsn vsn (u + v).
For if we take places xlt ..., £,„ such that
these 3p places will be the zeros of a rational function which has alt ..., at, as poles, each
to the third order. This function is expressible in the form (My + NP)/P2, where P
denotes (x — ax) ...... (x — ap\ Mis an integral polynomial in x of order p — l, and A7 is an
integral polynomial in x of order p. Denoting this function by Zt we have
dl 1
= ay>
217] EXAMPLES. 327
[X T^f/T
by § 154, Chap. VIII., where 1=1% '* = iXa> + , I - . Writing Z in the form
.' M y
and taking X2p + 1 = 4, we find the value of the integral K to be - 2A.
But from the equation
N*P-4JPQ = (x-xJ ...... (x-xp)(x-zl} ...... (#-*p)(A'-fi) ...... (*-W,
where Q = (x — c) (x — q) ...... (# - cp), we have, putting «; for x,
= 2 V - Q (<j (Aa*-1 + ...), (i = l, 2, ...,
where pi=\/(ai-xl) ...... («»-#,,), ft—^(«r-*i) ...... K-^), ^i = V(a»-fi) ...... («»-&>);
solving these equations for A we eventually have*
= 5
Ex. x. Obtain, for jo = 2, the corresponding expression for ^ (u~) + f j (v) -
Ex. xi. Denoting - 7-= — =. by (7v, the equation
'
= 2
gives
W= 2 <7i[pSr)fc-.Ptfir)] OT» (r=l, 2, ..., p),
where p/ denotes x— V(«i — *'i) ...... («i - ^P)- It has been shewn that pt is a single valued
function of u and it may be denoted by pi (u). Similarly Wi is a single valued function
of u + v, being equal to Pi( — u — v). The equation here obtained enables us therefore to
express pi(u+v] in terms of Pi(u], Pi(v), and the diflferential coefficients of these; for
we have obtained sufficient equations to express |ppi ,. (u\ ^>;)> r (v) in terms of the functions
Pi (u\ pi (v). A developed result is obtained below in the case p = 2, in a more elementary
way.
217. We have obtained in the last chapter (§ 197) the equation
S% j (ux' m - u*» "> - ...... - U*P> «P) in (x) & (xp) = DxDXpRxx'p°c.
i j
Hence, adopting that determination of the matrix a, occurring in the
integrals L*' *, and the function S- (u) (§ 192, Chap. X.), which gives the
particular forms for L*' * obtained in § 138, Ex. i., we have in the hyperellip-
tic case
(M*. « + w*.« •» + .. ..+U*P> tyaf-W-1
4 (a? - serf
P+I
where f(x, z) = S xlzl [2^ + \^+1 (x + z}\ This equation is, however, in-
t = 0
* This equation, with the integrals Lx< a on the left-hand side, is given by Forsyth, Phil.
Trans. 1883, Part i.
328 RATIONAL EXPRESSION [217
dependent of the particular matrix a adopted. For suppose, instead of the
particular integral
Tx,n (xdx^~i
Li , = — 2, Xjfc+i+i (k+l-i) &,
J H y k = i
we take
TX> * v n x> *
IH - 2, Ci>kuk ,
k=l
where Cijk = Ckji; then (§ 138) this is equivalent to replacing the particular
matrix a by a + \ C, where C is an arbitrary symmetrical matrix, and we
have the following resulting changes (p. 315)
R%c (p. 194) becomes changed to Rx^ * - S2C^ kuf °% °, so that,
f(x, z) (p. 195) becomes changed to/(#, z) - 4 (x — z)- 22(7;, kO?-1?*-1,
S-(w) (§ 189) becomes multiplied by
and thus ^(u) is increased by Cit 1itl + ...... + Ciipup, and instead of $>ij(u)
we have ^ j (u) — Cti j.
Since now ux> a + ux*> a« = ux»> a + ux> a«, we have \p (p + 1) equations of the
form
where u= ux>a+ ux" a> + ...... +uxp'ap, r = Q, I, ..., p, and 6' = 0, 1, ..., p.
Hence, if e1} e2 denote any quantities we obtain by calculation
* -
here the matrix a is arbitrary, the polynomial f(xr, xs) being correspond
ingly chosen, and
Suppose now that f(x, z) =f(x, z) + 4 (# — ^)2 SS^i, j«r «« , where
i i
f(x, z) is the form obtained in Ex. viii. § 211 ; then we obtain
ss EM») - A J •!-'.*-• - '
and by Ex. x. § 211 this is equal to
_
4 (&1 - e# R («
4 (6l - e2)2 ^ (e2) 4 (e, -
217] OF THE FUNCTIONS |>. 329
and therefore
4 (e, - e,J R («,) 4 (e, - etf R (<?2) **(*- etf
This is a very general formula* ; in it the matrix a is arbitrary.
It follows from Ex. xi. § 211 that if blt b2 be any branch places, we have
where E is a certain constant (cf. §§ 213, 212). This equation is also inde
pendent of the determination of the matrix a.
By solving %p(p + l) equations of this form, wherein 61} 62 are in turn
taken to be every pair chosen from any p + 1 branch places, we can express
22%>i,j (w) e\ eJ2 as a linear function of \p (p + 1) squared theta quotients,
el, e2 being any quantities whatever.
By putting 62 at a, that is at infinity (first dividing by 6f ~1), and putting
x also at a, this becomes the formula already obtained (§ 216)
Ex. i. When jo = 1, taking the fundamental equation to be
the expression >-**-**-*,
p+i
/(o;,4 =2^[2X2i + X2i
and
Therefore, by the formula at the middle of page 328, taking the matrix a to have the
particular determination of § 138, Ex. i.,
* \x-xj '
this is a well-known result.
Ex. ii. When p = 2, we easily find
It is given by Bolza, Gottinger Nachrichten, 1894, p. 268.
330 FORMULAE FOR THE FUNCTIONS |jf>. [217
and thus the expression
i 2 i. 2 i2 s, 2
is equal to
, a?g) (^ - e^ (^ - e2)
4(a?-a?2)2
_
(#2 — a?) (a?2 - a?x) 4 (a: — a^)2
Herein the matrix a is perfectly general. Adopting the particular determination of
§ 138, Ex. i., we have, since the term in f(x, z) of highest degree in x is A2p + 1#p + 12p, =4ar3s2,
say, by putting the place x at a, that is at infinity, the result
Pi, 1 («) + («1 + 62) Pi, 2 00 + *1*2 £>2> 2 («) =
, — 2
where u=uXl' ai + ux-' a2.
^r. iii. Prove, for jo = 2, when the matrix a is as in § 138, Ex. i., that
).^
— —
where e^ e2 are any quantities, u = uXl' ai + ux*' tt2, and ^19 /a2 are as in § 216 (cf. § 213).
Ex. iv. From the formula, for£> = 2 (§§ 217, 216, 213),
where a1} cr2 are the branch places as before denoted, infer (§ 216, Ex. iii.) that
Pll (*0 - Pll ('0 + Pl2 W P22 («') - Pl2 («') P22 («) = ^T [?122 ~ ?'l22 - ?!2?2' 2 +
Ct-j — W2
Prove also that, for any value of u, and any position of x,
ff>u (M*. « + u) - pu (w) + ^>12 (M*- « + M) ^>22 (M) - g>22 (u*> «
. v. If 61} ..., 6P + 1 be any (p + 1) branch places, and e1? e2 any quantities whatever,
= (#-&i) ...... (^-&P+i), J/'(^) = (^-e1)(^-e2)(^-61) ...... (^-&p + 1), prove that
where the matrix a has a perfectly general value, r, s consist of every pair of different
numbers from the numbers 1, 2, ..., (p + 1), and Er, 8 are constants.
218. We conclude this chapter with some further details in regard to
the case p = 2, which will furnish a useful introduction to the problems of
future chapters of the present volume. We have in case p = 1 such a formula
as that expressed by the equation
<r(u + u') a-(u — u') . ,. , .
.•ooVoo -tw-tw>
we investigate now, in case p = 2, corresponding formulae for the functions
- u')
218]
CONSIDERATION OF THE CASE IN WHICH p IS TWO.
331
by division of the results we obtain a formula expressing the theta quotient
^(M+W' ub'a)+^(u+u'} by theta quotients of the arguments u, u' ; this formula
may be called the addition equation for the theta quotient ^ (u ub> °) -r- S- (u).
Though we shall in a future chapter obtain the result in another way, it will
be found that a certain interest attaches to the mode of proof employed here.
Determine the places aclt xz, a?/, xz so that
u =
then, in order to find where the function <& (ux"ai + ux*>a*
vanishes, regarded as a function of a^, we are to put
i>ai + uXi'>a*)
thus the places z1} z2 are positions of xl for which the determinant
= ~D ~/ir\ ' ~~D~T^\ t **it -'-
, x2 , 1
wherein P (x) denotes (x — a^) (x — a2), vanishes. By considerations analogous
to those of § 209 we therefore find, V denoting the determinant derived
from V by changing the sign of y±, y2,
— ii'\ VVPfr ^ P(r \ P(r '\ P(<r '\
/A V V JL li//| / -L I a/2 / J- \^1 ) -»- \*v-2 /
S-2 (u) ^2 (u') " (a?! - x^f (a?/ - a;2')2 (x-, - a?/) (^ - a?/) (ara - a;/) (a?., - <) '
where -4 is an absolute constant.
Now, ifrjl=y1/P (tfj), etc., we find by expansion and multiplication,
V V = (vi»?2 + 'hV)2 (#1 ~ *'i)2 (x ~ *? ~
and, if a = (x^ - x^) (x2' - x2), /3= (x-[ - x,2) (x% - Xj), a - j3= (x-{ - x2') (^\ - .t'2), this leads to
but, putting y2 = 4P (x) Q (x), =<i(x-al)(x- «2) (x - c) (x - ct) (x - c2), we have
16
332 ADDITION FORMULA. OF THE ELLIPTIC CASE [218
and this expression is equal to
+ Q*L(a-x)(a-x)(a-x')(a-x')\,
P'ttj
as may be proved in various ways ; now we have proved (§§ 208, 212, 213) that
(«j - ^) («! — 3?2) = + \/ - P' («j) § (ttj) gl2, (a2— ^j) («2 - #2) = ± V — P' i
and
where gl = 3 (u \ ua> ' a) -r 5 (tt), y2 = 5 (w | Ma2' a) -^ 5 (M), ^, 2 = 5 (M | wa' ' a + ?ttt2- °) -f. 5 (w) ; thus
P(ae^P(s^P(x^f(x^
as qi*q£q,zq«z= 7> ,, N „, >\4 ^ , we have
'
8) P fa') P (argQ
.
2
where however we have assumed that the sign to be attached to the quotient
is the same for the places x^ x2' as for the places xlt x.2. The product V - P' (GJ) § (at
V - P' («i) § («!> is, of course, here equal to - P' (o^) § (e^). Now,
P ' (ai) = (^ _ a2) = _ p' («2) .
thus we obtain
^ (M + w') ^ (w - ^t/)
^
the value of the constant multiplier, S-2, = [S- (O)]2, being determined by
putting u' = 0, in which case <?/, qs'} q^ 2 all vanish.
If in this formula we write v = u + ua" a + ua*' a in place of M, we obtain, from the
formulae
which are easy to verify from the formulae of § 190, Chap. X. and the table of
characteristics given in this chapter, that
=
^
and therefore
219] GENERALISED TO THE CASE IN WHICH p IS TWO. 333
where 3 (u) denotes 3 (u \ «0" a + ua*' a). But we can use the result of Ex. iv. § 217, to give
the right-hand side a still further form, namely
Further if u"1' a + ua*' a=$Qm> TO,, where m, m' consist of integers each either 0 or 1,
we find, by adding |iim, m, to u and u' and utilising the fact (§ 190) that
\m (u + u') = 2A j™ («) + 2X^TO (u'\
that
where v = u + £ Qm> m, , y' = u' + % flm> m, . It should be noticed that
Pi, i (v) = ~ g^.Q log •? ('« ; i TO, |w') ; hence
* j
this formula can be expressed so as to involve only a single function in the
form
J2/ijyU.2 ff (u -\- 1?) <T (u, — v)
where a- (u) denotes ^ (u ^ (•,•,))• and caij(u)= — W cr (u\ In
V \J-J- // •"»•/»' c)u ' nit • *
i j
Weierstrass's corresponding formula for p = l, the function <r(7^) is de
termined so that cr (u)/u = 1 when u = 0. To introduce the corresponding
conditions here would carry us further into detail. (See §§ 212, 213.)
Ex. Prove that if «3 denote any one of the branch places c, c15 c2, a = (o2-«3),
^ = («3-a1), y=(a1-a2), P1=-(a1-xl) (a-^-x^ etc., Pi' = (al-a:l") (a^-xj), etc., and
with similar notation for A', B', then the determinant A can be expressed in the form
where
- A'
In this form A can be immediately expressed in terms of theta quotients.
219. Consider, nextly, the function
334
ADDITION FORMULA FOR CERTAIN
[219
This is not a periodic function of u, u . Thus we take in the first place
the function
Put
^ (u) ^ (u\ua» a) ^ (u') % (u ua>< «) '
u = ux> > a> + ux*> "», u' = uxi' ai + u**'
then, as functions of x1} the zeros of ^ (u), *b(u\ua"a) respectively are a, x2
and Oj, a;2, the zeros of ^(u + u'\uai1 a) are found in the usual way to be zeros
of a rational function of the fifth order having af, a.2s as poles, and xz, a?/, #2'
as zeros; such a function of #j is AX/P (a^), where P (#1) = (xl — a^ (xl — a2) and
i/! (a^ - cO, x? , xl , 1 !
'?2 (^2 ^h )> «^2 > ^2 > •*•
%' (a?/ - ax), ar/2, a;/, 1
wherein ^ = yx/P (a?,), etc. ; the zeros of *b(u - w')> as a function of xlt are
similarly zeros of a function of the sixth order having a^, a23 as poles and
a, tf2, x±, x2' for its other zeros ; such a function of xl is A/P(a?i), where
A =
hence we find
^2
- 77j V, -
-772V, -773', a?/, 1
*t (u) ^ (u\ua> > a) ^ (u') ^ (u' | M°« • a)
= (7
! A (a^ — «2) (a?2 — q2) (a?!7 - a2) (#/ — a2)
#2)2 (a?! — a?/) (a;x — #/) (a;2 — a;/) (x2 — x£) (a;/ — a;2')2 '
wherein C is an absolute constant ; for it is immediately seen that the two
sides of this equation have the same poles and zeros.
We proceed to put the right-hand side into a particular form; for this purpose we
introduce certain notations; denote the quantities c, cl5 c2, which refer to the branch
places other than ax, a2 by a3, a4, «6 in any order; denote («i - x^) («j - #2) by pi,
(cti - x{] («j — x2') by p^ ; denote by TTJ, $ the expression
i f ^1 . ff2 1 1
L(^i - «i) (^i - «>) (*» ~ «i) («a - %) J *« - ^i '
and write jo^y for pipjiri, y, with a similar notation n'i, y, ?)'i, y ; also let P (x) = (.v- a:) (^ - a2)>
1i=yilp(xi)> etc-
Then, by regarding the expression
(„ V \ (n f '\ (ft — T '\
(a% — &%) ^c*2 — o-j ) ^«2 ^2 <
219] HYPERELLIPTIC FUNCTIONS OF THE FIRST ORDER. 335
as a function of «2, and putting it into partial fractions in the ordinary way, we find that
it is equal to
-1- (*/ -.<)* (x. - «3) iaz^azf^ + . _!_, K _ Xtf (x _ ^ (*i'-«0(«i'-«j?
~ "-- - 3'
using then the identities
- (*a - «s) (*/ - #2') = fo' - #2) to' - «3) - « ~ #a) (** ~ «s)>
to - «3) « - #2') = to' - a?2) to' - 03) - to' - a?2) (x{ - 03),
we are able to give the same expression the form
.
1 — 2
where 1 1;^ = (Xl - «3) (^ - a4) (^ - «.), etc. ; thus
= - («2 - a4) («2 - «5) (^' - ^2')2 (^/ - *z) (^2' ~ ^2) (xi ~ az) (x* ~ «s) —3—79
f*2 J?2"'
+ ^1 ^l' ~ ^2) W ~ #2) ((^2' ~ «2) (^2' ~ «3) (*'/ ~ «4) « - «5)
+ (•»/ - «2) (^i' - «3) (#2' - a4) (a:2' -
Now we have, by expansion,
^ = (J?i'?2+I?i'W) (^1-^2) (•^I'-^
and in the product AA there will be two kinds of terms
(i) - »h V (»7i - 172) y (*i - xj « +^2' - 2«!),
where y denotes to' -a?,) to'-a?2) to'-^Jto'-^g), there being four terms of this kind
obtainable from this by the interchange of the suffixes 1 and 2, and the interchange of
dashed and undashed letters,
(ii) 77, (^2' - Xl) to' - xj to - j?s) {,,'« (^' - «,) (^ - xtf + r,^ (^ - 04) « - .r2)2
-^2to-a,)to'-.r2')2},
there being three other terms similarly derivable from this one.
Consider now the expression
and, of this, consider only the terms
(a.-, - at) (og - o6
336 ADDITION FORMULA FOR CERTAIN [219
by substitution of the values for pls etc., and arrangement, we immediately find that these
terms are equal to
this expression, as we see by utilising an identity which was developed at the commence
ment of the investigation, is equal to
w , — SJT,
- g 1 - 2 - 2 -1 - 2 ~ 2) (^8 ~ -^2)
where K denotes
>7i th' 2 (^/ - «i) (xi ~ xtf + 12 2 (•< - «i) « - #2) - "722 (*a - «i) (^i' - ^2')2]
Comparing this form with the terms occurring in the expansion for AAj, we obtain the
result
Now we have (§§ 216, 213, 212) the formulae p?=ntf, {=±(ai-aj} 2 ; we
2t 9^ /*t ^y
shall therefore pui pt = Mfqi, Pi,j = -^i,jqi,j ; hence by the formula (p. 334) the quotient
"tt)^ (u - u')
is a certain constant multiple of the function
Also we have M2=m, N\tj= + /xiW/(a,--o>-), where m = ± V -/' («i) when i=l or 2,
and /xj=±\//'(ai) when i=3, 4, 5. Hence it is easy to prove that the fourth powers
of the quantities (a2 - «4) («2 - a5) M1M3N13, NnN^N^Mz are equal.
Hence we have
where ^4. is a certain constant, and e a certain fourth root of unity. The
value of e is determined by a subsequent formula.
220. The equation just obtained (§ 219) taken with a previous formula
gives the result
a»a) _ e (guggg/ -f gVsg/fr) -I- qi^q'^q'^ + q'nq^'q^q^
V2
221] HYPERELLIPTIC FUNCTIONS OF THE FIRST ORDER. 337
and limiting ourselves to one case, we may now take the places az, «4, as to
be, respectively, c1} c2, c, and introduce Weierstrass's theta functions;
defining* the ten even functions ^s(u), ^.23(u), ...,^03(u) to be respectively
identical with the functions ^(u), ^ac(u), ..., ^cai(u), and the six odd functions
X_> (u), . . ., ^3 (u) to be respectively the negatives of the functions ^(MI (u), ..., ^-CCi (u),
the right-hand side of the equation is equivalent to
- /(X. <\. <X> Ci.' i CW CL. CL \ i (X. C^. <y <\ ' -U V V *V ^V
fc ^ rJg JQ2 ~01~ 12 i '•'S ~ 0-2 ~01 rJ12/ i '-'04 '-'24 « 14 *->3 ~T ~ 04 ~ 24 '•'H '•*3 .
here ^- denotes ^-(w), V denotes *b(u'), and (7 is an absolute constant.
This equation may be called the addition forrnula for the function ql} and is
one of a set which are the generalisation to the case p = 2 of such formulae
as that arising for p = 1 in the form
,. sn u en u' dn u' + sn u' en u dn u
l-frsn'ttBn'u'
By interchanging the suffixes 1 and 2 we obtain an analogous expression
for (b(u + u'\ua*>a)-7-'&(u + u'); if in this expression we add the half-period
ua»a to u we obtain an expression for the function *&(u + u ua"a + ua*'a)
-r-*b(ii + u'\uai> °); and if this be multiplied by the expression just developed
for the function ^(u +u'\uai> a) -r- S- (u + u) we obtain an expression for
S- (u + u uai-a + ua*> a) -r- ^f (u + u'), and it can be shewn that the form obtained
can be reduced to have the same denominator as in the expression here
developed at length. The formulae are however particular cases of results
obtained in subsequent chapters, and will not be further developed here.
For that development such results as those contained in the following
examples are necessary; these results are generalisations of such formulae
as sn (u + K} = en w/dn u which occur in the case p = 1.
Ex. Prove, if qt (u) = 3(u\uai> a) + 9 (u\ qitj(u) = 3(u\uai'a + uai>a) + 3(u\ etc., that
(see the table § 204, and the formulae Chap. X. § 190)
f/i
and obtain the complete set of formulae.
221 . In case p = 2 there are five quotients of the form S- (u\ ub< a) -=- ^ (u),
and ten of the form ^(u\ub>< a+ ub*<a) + S-(w), wherein b, blt b.2 denote any
finite branch places. Since the arguments u may be written in the form
ux» a> + ux*>a*, the fifteen quotients are connected by thirteen algebraic
relations. In virtue of the algebraic expression of these fifteen quotients,
they may be studied independently of the theta functions. We therefore
give below some examples of the equations connecting them.
* Konigsberger, Crelle, LXIV. (1865), p. 22. In the letter notation (§ 204) the reduced charac
teristic symbols are such (§ 203) that each of k,, k', is positive, or zero, and less than 2. In
Weierstrass's notation the reduced symbols have the elements A;', positive, or zero, and the elements
k, negative, or zero.
B. 22
338 FORMULAE CONNECTING THE THETA FUNCTIONS [221
Ex. i. There is one relation, known as Gopel's biquadratic relation, which is of
importance in itself, in view of developments that have arisen from it, and is of some
historical interest.
I (±4-
be three functions whose suffixes, together, involve all the five finite branch places. Then
these three functions satisfy a biquadratic relation, which, if the functions be regarded as
Cartesian coordinates in a space of three dimensions, represents a quartic surface with
sixteen nodal points.
In fact, if pa denote \/(« - #1) (« — #2)* an<^ Pb b denote the function
we have
where 6U 62, elt e2, es are the finite branch places in any order ; and if this be denoted by
it is immediately obvious that ^ (^, ^) = 2^2, =2/(.r), say, and x- ^(.r, 2) = ^— ; tnus
there is (§ 211, Ex. vii.) an equation of the form
where /(^1} ^2) ^s a certain symmetrical expression of frequent occurrence (cf. § 217), the
same whatever branch places 6n 62 may be, and A, B, C are such that ^ (#x, #2) vanishes
when for xlt x2 are put any one of the four pairs of values (6lt &2), (e2, e3), (e3, ej, (elt ez) ;
therefore the difference between any two expressions such as y?b b , formed for different
pairs of finite branch places, is expressible in the form Z#1#2 + M (xl+.v^) + N ; thus there
must be an equation of the form
where X, /*, v, p are independent of the places #u xz.
Similarly
But also it can be verified that
Pa, , a,PCl , c, -Pa, , <A, c,
thus we have
0,
&, 0, + W. c.+ '^c + P] l>'lf «, + /•>„ c2 + ">c + P'] = bai> aA , c2-^l2'
and when the expressions^ a , etc., are replaced by the functions qa a , etc. (§ 210), this
is the biquadratic relation in question. This proof is practically that given by Gopel
Crelle xxxv. 1847 . 291.
(Crelle, xxxv. 1847, p. 291).
221] WHEN p IS EQUAL TO TWO, 339
En: ii. Prove that
2 2
alt a2~Palt c, . 2
~ a _p—'+Pa
t*o v| l
2 2
(«i - «i) («i - c) (cx - a^ (Cl - c) (c - a,) (c - GI
and hence develop the method of Ex. i. in detail.
*l
Ex. iii. For any value of p prove
(a) that the squares of any p
ected by a linear relation,
(/3) that the squares of any p of the theta quotients
(a) that the squares of any p of the theta quotients qb, = 9 (u\ub-a) + 3(u), are
connected by a linear relation,
are connected by a linear relation. (Weierstrass, Math. Werke, vol. i. p. 332.) These
equations generalise the relations of Ex. ii.
Ex. iv. Another method of obtaining the biquadratic relations is as follows ; if
$ (v) =^e2niv(n
Tr=|v, and, in Weierstrass's notation,
so that x : y : z : t = l : qa^ c^ : qa^ Ci : qc, and if a, b, c, d denote the values of M, y, z, t
when v=0, and the linear function cx + dy-az-bt be denoted by (c, d, -a, -b), etc.,
then it can be proved, by actual multiplication of the series, that
e32 ( V) = (c, d,-a,- b), 9142 ( T) = (d, - c, - b, a), e0./ ( F) = (b, -a,d,- c)
Q^(V) = (a,b,c,d) , e* ( V) = (b,-a,- d,c\ e^(V) = (a,b,-c,-cr).
Relations of this character are actually obtained by Gopel, in this way. It will be
sufficient, for the purpose of introducing the subject of a subsequent chapter, if the
method of obtaining one of these relations be explained here. The general term of the
series e^ ( F) is (cf. the table § 204 and § 220)
where g' = £(l, °), 9' = i(lJ 0), namely is
_ e"i Oi (Mi+i)+«2w2] +Ji« [TU
thus the exponent of the general term in the product e^2 ( V) is niL, where L is equal to
7rt1 + ^)m2]
+ m^ + ni+mi + l
22—2
340 FORMULAE CONNECTING THE THETA FUNCTIONS [221
there are therefore four kinds of terms in the product according to the evenness or
oddness of the two integers n^-\-m^ «a + m2. Consider only one kind, namely when
ni+mi, n<t + m-2 are both even, respectively equal to <2.Nl, 2JV2, say; then L is equal to
•e t.n\~m\ if n2 — m2 if
if now we put -1 -- - -l = Mlt * =M2, we have
thus, to any assigned values of the integers Nlt JV2, Mlt M2 there correspond integers
MU n2, mlt m2 such that n^m^ n2 + m2 are both even ; therefore, as
is a term of the series 3 (v ; £ ( ) J , that is, of B01 (v), and
is a term of the series 5 (o ; | ^^ , that is, of 36 («;), and e^<2^'+1' = - l, it follows that
the terms of e022 ( V) which are of the kind under consideration consist of all the terms of
the product - 55 . 501 (v), or - ay. It can similarly be seen that the three other sorts of
terms, when Mj + wij is even and n2 + m2 odd, when nl+ml is odd and »2+m2 odd or even,
are, in their aggregate the terms of the sum bx+dz — ct.
We can also, in a similar way, prove the equations
©0362363 ( V) 014 ( V) + 000,002 ( V) Q. ( V) - 012001 0, ( V) 034 ( F),
0032 = 2 (ac - bd), 0232 = 2 (ad + be), 022 = 2 (a6 - erf), 0012 = 2 (afe + erf),
©03 denoting 003 (0), etc.
Hence the equation of the quartic surface is obtainable in the form
V2 (ac - bd) (ad+bc) (c, rf, - a, - b) (rf, -c, - 6~a)
+ V(a2 - 62 - c2 + rf2) (ab - erf) (6, - a, rf, - c) (a, b, c, rf)
, -a, -d,c)(a,b, -c, -rf).
A relation of this form is rationalised by Cayley in Crelle's Journal, LXXXIII. (1877),
p. 215. The form obtained is shewn by Borchardt, Crelle, LXXXIII. (1877), p. 239, to be the
same as that obtained by Gopel. See also Kummer, Berlin. Monats. 1864, p. 246, and
Berlin. Abhand. 1866, p. 64 ; Cayley, Crelle, LXXXIV., xciv. ; and Humbert, Liouville, 4me Ser.,
t. ix. (1893); Schottky, Crelle, cv. pp. 233, 269; Wirtinger, Untersuchungen iiber Theta-
functionen (Leipzig, 1895).
The rationalised form of the equation, from which the presence of the sixteen nodes is
obvious, is obtained in chapter XV. of the present volume.
221] WHEN p IS EQUAL TO TWO. 341
Ex. v. Obtain the following relations, connecting the ratios of the values of the even
theta functions for zero values of the arguments when p = 2. They may be obtained from
the relations (§ 212)
(&-*,) (6-*2) = ±*Je"iPt"f> (b) V (U\ub- *)+& (M)
by substituting special values for xl and x2.
54 : 3* : 3* : 34 : 34 : 44 • 34 -A4 • Q4 . a4
c c, "c2 "a^e, ^a,Ca • ^a2Cl • ^a^ • ^ . 9^
= fa - c2) (eg - c) (c - Cl) . («! - Og) : (G^ - a2) («2 - c) (c - a,) . (c, - c2)
: (a, - og) (ctg - Cj) (cx - a,) . (c2 - c) : (ax - a2) (a2 - c2) (c2 - ax) . (Cl - c)
: (eg - c) (c - ax) (a, - c2) . (c, - «2) : (c - GI) (cx - a,) (^ - c) . (c2 - «2)
: (c - cg) (eg - og) (ag - c) . (ct - aj : (c - cx) (GI - «2) («2 - c) . (c2 - ttl)
• («l-*j(e»-*d(*»-«d-(*i-o) • (C1-c2)(c2-a1)(a1-c1).(a2-c).
Infer that
5^U : <A : ^< = K-ci)2 : ("i-«,)2 : («!-«,)'.
We have proved (§§ 210, 213) that
-a ^c M 5aa M = 0
and we have in fact, as follows from formulae developed subsequently, the equation
3caA2«2-^, fa) Sa,Cl (M) +^lC25cai5a2 (M) 9aic, (u) = 3c3c^Cl (u) 5a,a2 (i«).
Ex. vi. Obtain formulae to express the ratios of the differential coefficients of the odd
theta functions for zero values of the arguments.
Ex. vii. Prove that
(u) = f^
wherein blt 62 are any two finite branch places, and e is a certain fourth root of unity.
This result can be obtained . in various ways ; one way is as follows : Writing
M = M*.. «. + #*..«., u + ub»a=v, and v = uZl>b> + uz*'b*, we find, by the formula 3(
= e^(u)3(u; P), that
and, by the formula expressing & (ux> m-uXl'm<- ...... - uxi" *"») - fa (u1*' m - ux* 'm>-
- UXP< »"P) by integrals and rational functions, the right-hand side is equal to
Sl _s2_ -i
- &i) («i - 68) («g - 6,) & - 6g) J '
where sly zl are the values of y, x respectively at the place zly and s2, z2 at the place z2.'
This rational function of «,, 22 is however (§ 210) a certain constant multiple of
3 (v\ubt ' a + ub» a)/3 (v), and hence the result can immediately be deduced.
One case of the relation, when b19 62 are the places a,, a2, is expressible by Weierstrass's
notation in the form
a_
342 FORMULA FOR DIFFERENTIAL COEFFICIENTS. [221
and it is interesting, using results which belong to the later part of this volume, to
compare this with other methods of proof. We have*
-v) = 3S (u) 5M (tt) $i (v) 30 (v) + 32 (u) 513
(tt) + 32 (t;) 5U (*) 502 (tt) 5M (tt),
where 54, 50 denote 54(0), 50(0), and the bar denotes an odd function; if, herein, the
arguments vlt vz be taken very small, we may write 5 (u + v) = 5 (tt) + ( vl = — h #2 ~— ) 5 (u).
Thus we obtain, eventually, remembering that the odd functions, and the first differential
coefficients of the even functions, vanish for zero values of the arguments,
where y(u) = ^—3(u\ 5 = 5(0), 5' = 5'(0).
d«2
Thus, by the formula of this example, putting u = 0, we infer that
-0
or 3'o4 = 0, and the result of the general formula agrees with the formula of this example.
In the cases />>2 we have even theta functions vanishing for zero values of the
argument ; here we have one of the differential coefficients of an odd function vanishing
for zero values of the argument.
Note. Beside the references given in this chapter there is a paper by Bolza,
American Journal, xvn. 11 (1895), "On the first and second derivatives of hyper-
elliptic (r-functions " (see A eta Math. xx. (Feb. 1896), p. 1 : "Zur Lehre von den hyper-
elliptischen Integralen, von Paul Epstein"), which was overlooked till the chapter was
completed. The fundamental formula of Klein, utilised by Bolza, is developed, in
what appeared to be its proper place, in chapter XIV. of the present volume. See also
Wiltheiss, Crelle, xcix. p. 247, Math. Annal. xxxi. p. 417; Brioschi, Rend. d. Ace. dei
Lincei, (Rome), 1886, p. 199; and further, Konigsberger, Crelle, LXV. (1866), p. 342;
Frobenius, Crelle, LXXXIX. (1880), p. 206.
To the note on p. 301 should be added the references ; Prym, Zur Theorie der
Functnen. in einer zweibldtt. FUicJie (Ziirich, 1866), p. 12; Konigsberger, Crelle, LXIV. p. 20.
To the note on p. 296 should be added; Harkness and Morley, Theory of Functions,
chapter vin., on double theta functions. In connection with § 205, notations for theta
functions of three variables are given by Cayley and Borchardt, Crelle, LXXXVII. (1878).
* Krause, Hyperelliptische Functional, p. 44 ; Konigsberger, Crelle, LXIV. p. 28.
223]
CHAPTER XII.
A PARTICULAR FORM OF FUNDAMENTAL SURFACE.
222. JACOBI'S inversion theorem, and the resulting theta functions, with
which we have been concerned in the three preceding chapters, may be
regarded as introducing a method for the change of the independent variables
upon which the fundamental algebraic equation, and the functions associated
therewith, depend. The theta functions, once obtained, may be considered
independently of the fundamental algebraic equation, and as introductory to
the general theory of multiply-periodic functions of several variables ; the
theory is resumed from this point of view in chapter XV., and the reader
who wishes may pass at once to that chapter. But there are several further
matters of which it is proper to give some account here. The present chapter
deals with a particular case of a theory which is historically a development*
of the theory of this volume ; it is shewn that on a surface which is in many
ways simpler than a Riemann surface, functions can be constructed entirely
analogous to the functions existing on a Riemann surface. The suggestion is
that there exists a conformal representation of a Riemann surface upon such
a surface as that here considered, which would then furnish an effective
change of the independent variables of the Riemann surface. We do not
however at present undertake the justification of that suggestion, nor do
we assume any familiarity with the general theory referred to. The present
particular case has the historical interest that in it a function has arisen,
which we may call the Schottky-Klein prime function, which is of great
importance for any Riemann surface.
223. Let a, /3, 7, 8 be any quantities whatever, whereof three are
definitely assigned, and the fourth thence determined by the relation
aS — /3y = l. Let £, £' be two corresponding complex variables associated
together by the relation f = (a£ + /3)/(y£ + 8). This relation can be put into
the form
Referred to by Riemann himself, Ges. Werke (Leipzig, 1876), p. 413.
344
FORMATION OF A GROUP OF SUBSTITUTIONS
[22:i
wherein p is real, and B, A are the roots of the quadratic equation
£=(«£" + /3)/(7£+ S), distinguished from one another by the condition that
fj, shall be less than unity. In all the linear substitutions which occur in
this chapter it is assumed that B, A are not equal, and that p is not equal to
unity. We introduce now the ordinary representation of complex quantities
by the points of a plane. Let the points A, B be marked as in the figure (6),
Fig. 6.
and a point C' be taken between A, B in such a way that 1 > AC'/C'B > p,
but otherwise arbitrarily ; then the locus of a point P such that AP/PB
= AC'/C'B is a circle. Take now a point C also between A and B, such that
CB/A C= pC'SfAC', and mark the circle which is the locus of a point P'
for which P'B(AP'=CB/AC; since P'B/AP' is less than unity, this circle
will lie entirely without the other circle. If now any circle through the
points A, B cut the first circle, which we shall call the circle C', in the points
P, Q, and cut the second circle, C, in Px and Qj , P and Pj being on the same
side of AB, we have angle AP^B = angle APB, and P1B/APl = pPB/AP ;
therefore, if the point P be £, and the point Pj be £, we have
the argument of P vanishing when P is at the end of the diameter of the
C' circle remote from C", and varying from 0 to 2?r as P describes the circle
C' in a clockwise direction ; if then we pass along the circle C in a counter
clockwise direction to a point P' such that the sum of the necessary positive
rotation of the line BP1 about B into the position BP', and the necessary
negative rotation of the line APl about A into the position AP', is K, and f
be the point P', we have
- /,I
~
Thus the transformation under consideration transforms any point £ on
the circle C' into a point on the circle C. If £ denote any point within C'
224] IN CONNECTION WITH 2p CIRCLES. 345
the modulus of (£- B)/(£ — A) is greater than when £ is on the circumference
of G', and the transformed point £" is without the circle C, though not
necessarily without the circle C'. If £ denote any point without G' the
transformed point is within the circle G.
224. Suppose * now we have given p such transformations as have been
described, depending therefore on 3p given complex quantities, whereof 3 can
be given arbitrary values by a suitable transformation z' = (Pz + Q)/(Rz + $)
applied to the whole plane ; denote the general one by
. wherein o^ - &••/< = !> (i = 1, 2, . . . , p),
or also by
the quantities corresponding to A, B, /A, a being denoted by Aiy Bi, fj,i} cti ;
construct as here a pair of circles corresponding to each substitution, and
assume that the constants are such that, of the 2p circles obtained, each is
exterior to all the others ; let the region exterior to all the circles be denoted
by S, and the region derivable therefrom by the substitution ^ be denoted
by *&
If the whole plane exterior to the circle Ci be subjected to the trans
formation S-;, the circle C{ will be transformed into Cit the circle Ct itself
will be transformed into a circle interior to Ci} which we denote by ^(7;, and
the other 2p — 2 circles which lie in a space bounded by Ct and C{ will be
transformed into circles lying in the region bounded by ^iCt and Cit and,
corresponding to the region S, exterior to all the 2p circles, we shall have a
region ^S also bounded by 2p circles. But suppose that before we thus
transform the whole plane by the transformation ^, we had transformed
the whole plane by another transformation ^ and so obtained, within Gj,
a region %$ bounded by 2p circles, of which Gj is one. Then, in the
subsequent transformation, S-f , all the 2p — l circles lying within Gj will be
transformed, along with Cj, into 2p - 1 other circles lying in a region, ^fyS,
bounded by the circle ^Cj. They will therefore be transformed into circles
lying within ^(7,- — they cannot lie without this circle, namely in S-;$, because
*&iS is the picture of a space, S, whose only boundaries are the 2p funda
mental circles Clt CV, ..., Cp, Gp'. Proceeding in the manner thus indicated
we shall obtain by induction the result enunciated in the following statement,
wherein S^ is the inverse transformation to ^, and transforms the circle Gt
into Ci': Let all possible multiples of powers of^l} S-f , ..., *&p, \l be formed,
and the corresponding regions, obtained by applying to S the transformations
* The subject-matter of this section is given by Schottky, Crelle, ci. (1887), p. 227, and
by Burnside, Proc. London Math. Soc. xxni. (1891), p. 49.
346
DIAGRAM TO ILLUSTRATE THE RELATIONS
Fig. 7.
[224
5*
5
DD
»•*
DD
D
an
D
n
an
a
an
a
DD
D
an
an
a
S-^-i
fi-a J
•J "«p
an
D
Inn
Sr'f'3
<
&*$-$
na
an
n
DD
3
an
a
nn
a
ftty*
a
an
a
an
aa
a
&<!>*
n
an
3'/d>3"/
$4*
a
an
an
a
an
a
an
a
an
a
«
a
aa
a
an
»4f
an
n
a
an
pSr*
t-1
V
(Cl
«J,-'S-'
n
na
an
a
an
a
9
«!•
'•3
an
a
nn
a
n
an
n
nn
D
an
n
na
a
an
n
nn
a
an
a
aa
na
n
•fr'Sty
/
•fr-'S-ty
na
a
nn
n
<j>-'S<f>-J
<j>-^<f)
aa
a
n
nn
psfr*
j ^<9&
9 "-o
a
aa
nn
n
an
n
na
a
na
a
ril
•-J
n
an
n
an
x~
nn
n
«f>-3
t
224]
BETWEEN THE SUBSTITUTIONS AND THE REGIONS.
Fig. 7.
347
3 ,:«;
*
y
D
DD
DD
D
DD
D
DD
D
DD
D
DD
D
DD
DD
D
W
3-4
DD
D
DD
ty-'fr'
*p*
D
DD
a
DD
a*a*
d*fi
D
DD
DD
D
DD
D
DD
D
DD
D
DD
D
DD
n
DD
D
da
n
„-
*
D
DD
D
DD
D
DD
D
DD
DD
D
af*
s^
DD
D
D
DD
<|>S-2
J.A
-j
«j»s2
n
an
DD
D
DD
D
4
)O
an
a
an
n
0
DD
D
DD
D
an
a
an
a
an
a
an
a
an
n
an
nn
4>^''<|>-
<j>s-^
an
D
an
a
W7
4>3<J>
an
a
a
an
#t$ri
<j.2d
a
an
an
a
DD
a
an
a
an
D
!>
,4
•
a
DD
n
an
V
an
a
348 CHARACTERISTICS OF THE GROUP. [224
corresponding to all such products of powers, be marked out. In any such
product the transformation first to be applied is that one which stands to the
right. Let m be any one such product, of the form
m= ...... /*,
formed by
...... + rt + rj +rk, =h
factors, and let ^ be any transformation other than the inverse of ^, so that
m^fk is formed by the product ofh+l, not h — I, factors. Then the region mS
entirely surrounds the region m^S.
Thus, the region $r{$ entirely surrounds the space ^i&jS, and the latter
surrounds $•£•/$, or ^ib/b^S ; but ^ftS is surrounded by ^f^^S or S. The
reader may gain further clearness on this point by consulting the figure (7),
wherein, for economy of space, rectangles are drawn in place of circles, and
the case of only two fundamental substitutions, S-, <f>, is taken.
The consequence of the previous result is — The group of substitutions
consisting of the products of positive and negative powers of S-x , . . . , % gives
rise to a single covering of the whole plane, every point being as nearly reached
as we desire, by taking a sufficient number of factors, and no point being
reached by two substitutions.
225. There are in fact certain points which are not reached as trans
formations of points of S, by taking the product of any finite number of
substitutions. For instance the substitution ^m is
% ~ ^i %~ •"*
and thus when m is increased indefinitely £' approaches indefinitely near to
BI , whatever be the position of £ ; but Bt is not reached for any finite value
of m. In general the result of any infinite series of successive substitutions,
K = a/37 . . . , applied to the region S, is, by what has been proved, a region
lying within 08, in fact lying within a/3S, nay more, lying within a@yS, and
so on — namely is a region which may be regarded as a point ; denoting it by
K, the substitution K transforms every point of the region 8 and in fact
every other point of the plane into the same point K ; and transforms the
point K into itself. There will similarly be a point K' arising by the same
infinite series of substitutions taken in the reverse order.
Such points are called the singular points of the group. There is an
infinite number of them ; but two of them for which the corresponding
products of the symbols ^ agree to a sufficient number of the left-hand
factors are practically indistinguishable ; none of them lie within regions that
are obtained from S with a finite number of substitutions. The most
important of these singular points are those for which the corresponding
226] COMPOSITION OF TWO SUBSTITUTIONS. 849
scries of substitutions is periodic ; of these the most obvious are those formed
by indefinite repetition of one of the fundamental substitutions ; we have
already introduced the notation
to represent the results of such substitutions.
226. If S-, <f> be any two substitutions given respectively by
„
8'
wherein <z8 — /3y —- 1 = AD — BC, the compound substitution ^<£ is given by
D) (7A + BC) Z+(yB + SD)'
and if this be represented by £" = («'£+ P')/(y'£+ 8'), we have, in the ordinary
notation of matrices
( a! £') = (« P ) ( A B ),
7
C D
and of 8' - fi'y = (08 - £7) (AD- BC) = 1. We suppose all possible substitu
tions arising by products of positive and negative powers of the fundamental
substitutions S-15 ..., *&p to be formed, and denote any general substitution by
£' = (&£+ /3)/(y£+ 8), wherein, by the hypothesis in regard to the funda
mental substitutions, a& — fiy = l. We may suppose all the substitutions
thus arising to be arranged in order, there being first the identical substitution
£' = (f+ 0)/(0. £+ 1), then the 2p substitutions whose products contain one
factor, ^ or ^r1, then the 2p(2p-l) substitutions whose products are of
one of the forms ^^-, S^-1, ^r1^, ^r1^"1, in which the two substitutions
must not be inverse, containing two factors, then the 2p (2p — I)2 substitutions
whose products contain three factors, and so on. So arranged consider the
series
2 (mod 7)"*,
wherein & is a real positive quantity, and the series extends to every sub
stitution of the group except the identical substitution. Since the inverse
substitution to ? = (*S+/3)/(rt+B) is £=(B? - £)/(- 7f +a), each set of
2p (2p- I)"-1 terms corresponding to products of n substitutions will contain
each of its terms twice over.
Let now ®n denote a substitution formed by the product of n factors,
and @n+1 = @n^t-, where ^ denotes any one of the primary 2p substitutions
^1,^1 , -.., \, % other than the inverse of the substitution whose symbol
stands at the right hand of the symbol @n, so that @n+1 is formed with n + 1
350 POINCARE'S INFINITE SERIES [226
factors; then by the formula just set down <yn+1 = yn<Xi + Sny{, where, if
*t-, or £' =(OiC+ &)/(7if+8i), be put in the form (£' - Bt)/(? - A{)
= Pi(£- Bi)l(t;-Ai), we have
respectively equal to
Bjpj -Ajpi AjBj(pi —p^ pi -pi Ajpj
the signification of p^ is not determined when the corresponding pair of
circles is given ; but we have supposed that the values of Of, &•, 7;, Si are
he value of pt . By these formulae
7n+i _ _ &/fo _ AJ + 8nfyn
~
given, and thereby the value of pt . By these formulae we have
Herein the modulus of pi may be either fn or /if1, according as S; is one
of Si, ..., Sp or one of Si"1, • ••> S^1 ; the modulus of pi is accordingly either
less or greater than unity. If now ®n = ... >/r<£Sr \ where S> is one of the
2p fundamental substitutions S1( ...,^fpl, and therefore @^1 = Sr^>~1/^~1...,
the region ®nlS lies entirely within the region S>$ (§ 224) or coincides with
it; wherefore the point ©^(oo ), or — Sn/Vn> lies within the circle Cr when
S> is one of ^1( ..., *&p, and lies within the circle Gr' when X is one of
Si"1, ... , S^1; thus the points Bi and —Sn/yn can only lie within the same
one of the 2p fundamental circles C1} ..., C^,' when r=t and S> is one of
Si, ...,Sy, and the points Ai and —Snfyn can only lie within the same one of
the 2p fundamental circles Clt ..., Cp' when r=i and Sr is one of S-f1, ...,*bpl.
Now, if the modulus of p{ be less than unity, and r = i, ^fr must be one
of Sf , ..., S-p , namely must be Sr1, since otherwise @nSi would consist
g
of n — 1 factors, and not n + 1 factors ; in that case therefore Bi + —
Vn
is not of infinitely small modulus ; if, however, the modulus of p^ be
greater than unity, and r = i, S> must be ^, namely one of Si, ..., Sp, and
in that case the modulus of A{ + Sn/Yn is not infinitely small. Thus, according
pi ^1, we may put
as
I Bi + Bnfvn \>\, I Ai + 8n/
where X is a positive real quantity which is certainly not less than the
distance of Bi} Ai} respectively, from the nearest point of the circle within
which — 8n/yn lies.
227] ASSOCIATED WITH THE GROUP. 351
It follows from this that we have
mod (7n+1/7«) > <r, or mod (y~ljy~l) < - .
where a- is a positive finite quantity, for which an arbitrary lower limit may
be assigned independent of the substitutions of which ©„ is compounded, and
independent of n, provided the moduli fj,1} ..., fj,p be supposed sufficiently small,
and the p pairs of circles be sufficiently distant from one another.
Ex. Prove, in § 223, that if C' be chosen so that C'C is as great as possible
J_ C"<7=1-V^_1
fa Ati~l+fa fa
and the circles are both of radius c£ V/*/(l -/*), where d is the length of AB.
We suppose the necessary conditions to be satisfied ; then if j0 be the
least of the p quantities mod [(/*rVJi"< - ^.^/(Bi-Ai)], and k be posi
tive, the series 2 mod 7"* is less than
\f) , 2jp(2p-l) 2»(2«-l)2 1
2^ + -^ l ' + * — L+ ...... ,
|_ * a* o-2*
_) ,
70 ^ + -
|_ *
and therefore certainly convergent if ck > ~2p - 1, which, as shewn above, may
be supposed, //1( ..., /j,p being sufficiently small.
227. Hence we can draw the following inference: Let a-1} ..., <rp be
assigned quantities, called multipliers, each of modulus unity, associated
respectively with the p fundamental substitutions ^, ..., \; with any
compound substitution VV2---, let the compound quantity trf*af*... be
associated: let f(x) denote any uniform function of x with only a finite
number of separated infinities; let f = (a£ + 0)/(y£ + 8) denote any sub
stitution of the group, and cr be the multiplier associated with this
substitution : then the series, extending to all the substitutions of the group,
converges absolutely and uniformly * for all positions of f other than (i) the
singular points of the group, and the points f=-g/7, namely the points
derivable from £= GO by the substitutions of the group, including the point
f =00 itself, (ii) the infinities of /(£) and the points thence derived by the
substitutions of the group. The series represents therefore a well-defined
continuous function of f for all the values of f other than the excepted ones.
The function will have poles at the poles of /(£) and the points thence
derived by the substitutions of the group; it may have essential singularities
at the singular points of the group and at the essential singularities of
In regard to f ; for the convergence was obtained independently of the value of
352 COMPARISON OF THE £ PLANE [227
Denote this function by F(£); if S-0 denote any assigned substitution
of the group, and ^ denote all the substitutions of the group in turn, it is
clear that ^0 denotes all the substitutions of the group in turn including the
identical substitution ; recognising this fact, and denoting the multiplier
associated with ^0 by <TO, we immediately find
or, the function is multiplied by the factor (TQ~l(^+ S0)k when the variable
£ is transformed by the substitution, &0, of the group. Thence also, if G(£)
denote a similar function to F(g), formed with the same value of k and
a different function /(£), the ratio F(£)IG(g) remains entirely unaltered
when the variable is transformed by the substitutions of the group. In order
to point out the significance of this result we introduce a representation
whereof the full justification is subsequent to the present investigation.
Let a Riemann surface be taken, on which the 2p period loops are cut ; let
the circumference of the circle Ct of the £ plane be associated with one side
of the period loop (6t-) of the second kind, and the circumference of the circle
Ci with the other side of this loop ; let an arbitrary curve which we shall
call the t'-th barrier be drawn in the £ plane from an arbitrary point P
of the circle (7/ to the corresponding point P' of the circle Git and let the
two sides of this curve be associated with the two sides of the period loop
(di) of the Riemann surface. Then the function F(%)/G(£), which has the
same value at any two near points on opposite sides of the barrier, and
has the same value at any point Q of the circle G{ as at the corresponding
point Q' of the circle C{, will correspond to a function uniform on the
undissected Riemann surface. In this representation the whole of the
Riemann surface corresponds to the region S ; any region ^S corresponds to
a repetition of the Riemann surface ; thus if the only essential singularities
of JP(£)/6r(f) be at the singular points of the group, none of which are
within 8, F(£)/G(£) corresponds to a rational function on the Riemann
surface. It will appear that the correspondence thus indicated extends to
the integrals of rational functions ; of such integrals not all the values can
be represented on the dissected Riemann surface, while on the undissected
surface they are not uniform ; for instance, of an integral of the first kind,
ut, the values in, Mi+2a>i>r, w; + 2&>'iir, m + 2a>f) r + 2a)'it r may be repre
sented, but in that case not the value u{ + 4<w,:) r ; in view of this fact the
repetition of the Riemann surface associated with the regions derived from
8 by the substitutions of the group is of especial interest — we are able to
represent more of the values of the integral in the % plane than on the
Riemann surface. These remarks will be clearer after what follows.
228. In what follows we consider only a simple case of the function
F(Q, that in which the multipliers alf ..., <rp are all unity, k = 2, and
/(£) = !/(£— a), a being a point which, for the sake of definiteness, we
228] WITH A RIEMANN SURFACE. 353
suppose to be in the region S, We denote by & = ^ (£) = (a£ + &)/(7i(T + ^<)
all the substitutions of the group, in turn, and call £f the analogue of £ by
the substitution in question. The function
has essential singularities at the singular points of the group, and has poles
at the places %=a, £= oo and at the analogues of these places. Let the
points oo , a be joined by an arbitrary barrier lying in S, and the analogues of
this barrier be drawn in the other regions. Then the integral of this
uniformly convergent series, from an arbitrary point £, namely, the series
is competent to represent a function of f which can only deviate from uniformity
when £ describes a contour enclosing more of the points a and its analogues
than of the points oo and its analogues ; this is prevented by the barriers.
Thus the function is uniform over the whole f plane; it is infinite at £=a
like log(f — a), and at £=oo like — log (-^ I, as we see by considering the
term of the series corresponding to the identical substitution ; its value on
one side of the barrier 0.00 is 2?™' greater than on the other side ; it has
analogous properties in the analogues of the points a, oo , and the barrier aoo ;
further, if £n = ^n(Z) be any of the fundamental substitutions S-j, ..., <&p,
where £in is obtained from £ by the substitution S^n ; since the first and
last of these sums contain the same terms, we have
and the right-hand side is independent of £, being equal to II «^f; in order
to prove this in another way, and obtain at the same time a result which
will subsequently be useful, we introduce an abbreviated notation ; denote
the substitution S> simply by the letter r; then if j be in turn every sub
stitution of the group whose product symbol has not a positive or negative
power of the substitution n at its right-hand end, all the substitutions of the
group have the symbol jnh, h being in turn equal to all positive and negative
integers (including zero) ; hence
2 [log (£» - a) - log (£ - a)], = 2 S [log (£M* * , - a) - log (&»* - a)],
i j h
is equal to
B. 23
I D R A FJ y~
OF THF
TT-XTTTT-T-^-n ^,-r --, ,
354 THE KIEMANN INTEGRALS [228
where N = nx, M = n"x> ; but, in fact, £v is Bn, and £„ is An ; thus H^"^ is
independent of £ ; and if we introduce the definition
where S-?l is one of the £> fundamental substitutions, and, as before, j denotes
all the substitutions whose product symbols have riot a power of n at the
right-hand end, we have
a, oo a,
If for abbreviation we put
prove that
pf«- f _ 2_ p£ £ = 7>c»> c
O., oo (Tn «. °° «, =0
c being an arbitrary point.
229. Introduce now the function Ua,b defined by the equation
then, because a cross ratio of four quantities is unaltered by the same linear
transformation applied to all the variables, we have also
_v. /^-f/a.-
- log
where r, denoting S>, =^~1, becomes in turn every substitution of the group.
Thus we have
where
• - 2 ~- „, 6 ,
j denoting as before every substitution whose product symbol has not a
positive or negative power of n at the right-hand end and £ being arbitrary ;
hence also
where ?*, = t"1, denotes every substitution of the group.
230] CONSIDERED ON THE £ PLANE. 355
There are essentially only p such functions v» °, according as ^n denotes
^i, ^2, ..., ^,; for, taking the expression given last but one, and putting
n = st, that is, ^n = &,&«, we have
'a *"'* — f»"^ *"*
- n v *" -4- n " *
1 &« H £« »
where r) = %t, so that
and in particular, when st is the identical substitution, as we see by the
formula itself,
n f , a . £ «
0 = vg + v> J
thus, if r denote SY &•/ . . . . . . , we obtain
V =
so that all the functions v^ a are expressible as linear functions of v\ , ..., Vp .
230. It follows from the formula
that the function vn' rt is never infinite save at the singular points of the
group. But it is not an uniform function of £; for let £ describe the circum
ference of the circle Cn in a counter clockwise direction ; then, by the factor
%—Bn, v^a increases by unity; and no other increase arises; for, when the
region within the circle Cn, constituted by *bnS and regions of the* form
^n</»S, contains a point S-j(5n), the product representing the substitution j has
a positive power of ^-n as its left-hand factor, and in that case the region
contains also the point §j(An). Similarly if £ describe the circle Cn' in a
clockwise direction, v^ increases by unity. But if % describe the circum
ference of any other of the 2p circles, no increase arises in the value of
Vn a, for the existence of a point Sj (Bn) in such a circle involves the existence
also of a point Sj (An).
It follows therefore that the function can be made uniform in the region
S by drawing the barrier, before described, from an arbitrary point P of Cn' to
the corresponding point P' of Cn. Then v^, a is greater by unity on one side
of this barrier than on the other side. Further if in denote any one of
the substitutions ^1( ..., *$rp, we have
* Where 0 denotes a product of substitutions in which ^~! is not the left-hand factor.
23—2
356 THE RIEMANN INTEGRALS [230
where £ is arbitrary; thus as nf'"'f = Ilf"')f, the difference is also indepen
dent of £ and we have, introducing a symbol for this constant difference,
It follows therefore that if the p barriers, connecting the pairs of circles
Cn', Cn, and their analogues for all the substitutions, be drawn in the
interiors of the circles, the functions Vi °, . . . , vp a are uniform in the region S,
and in all the regions derivable therefrom by the substitutions of the group.
The behaviour of the functions Vi'a, ..., vpa in the region S is therefore
entirely analogous to that of the Riemann normal integrals upon a Riemann
surface, the correspondence of the pair of circumferences Cn, Cn' and the two
sides of the barrier P'P, to the two sides of the period loops (bn), (an), on the
Riemann surface, being complete. And the regions within the circles
C\, ..., Cp enable us to represent, in an uniform manner, all the values of the
integrals which would arise on the Riemann surface if the period loops (bn)
were not present. Thus the £ plane has greater powers of representation
than the Riemann surface. Further it follows, by what has preceded, that
the integral IT0')6 is entirely analogous to the Riemann normal elementary
integral of the third kind which has been denoted by the same symbol in
considering the Riemann surface. On the Riemann surface the period loops
(an) are not wanted for this function, which appears as a particular case of a
more general canonical integral having symmetrical behaviour in regard to
the first and second kinds of period loops ; but the loops (bn) are necessary ;
they render the function uniform by preventing the introduction of all the
values of which the function is capable. In the £ plane, however*, the
function is uniform for all values of £, and the regions interior to the circles
enable us to represent all the values of which the function is susceptible.
Thus the introduction of Riemann's normal integrals appears a more natural
process in the case of the £ plane than in the case of the Riemann surface
itself.
231. We may obtain a product expression for Tn>m directly from the
formula
Cm -*/(£«) £» - % (4
let k denote in turn every substitution whose product symbol neither has a
power of ^rm at its left-hand end nor a power of S>, at its right-hand end ;
thus we may write d>«^» %», or, for abbreviation, j = m~hk ; and for every
substitution k, the substitution j has all the forms derivable by giving to h
all positive and negative integral values including zero, except that, when k
* Barriers being drawn to connect the infinities of the function.
232] OF THE THIRD AND FIRST KINDS. 357
is the identical substitution, if m = n, h can only have the one value zero ;
then applying S/1 to every quantity of the cross ratio under the logarithm
sign, we have
JL V locr (Zj-im — Bn I £j-im-^?A
' n. in — o • •*-• iu>i \ i* n ~it ~A
2-rri j 6 V ?j-i - Bn I £,-_! - An j
= — 2 IQ
and therefore, if m be not equal to n,
r .
V*) - Bj **\Am) - Aj '
while when m = •/?., separating away the term for which k is the identical
substitution,
1
— 2' log
* *
where 2' denotes that the identical substitution, ^ = 1, is not included ;
thus
where s denotes every substitution of the group other than the identical
substitution, not beginning or ending with a power of X, and excluding
every substitution of which the inverse has already occurred.
These formulae, like that for wj a, are not definite unless the barriers (§ 227)
are drawn.
232. Ex. i. If v,,' = un + iwn , un, iion being the real and imaginary parts of v% a, prove,
as in the case of a Riemann surface, by taking the integral fu dw round the p closed
curves each formed by the circumferences of a pair of circles and the two sides of the
barrier joining them, that the imaginary part of N*TU + ...... + 2A\Nzrl2 + ...... is positive,
Ni , . . . , Nv being any real quantities, and u + iw = Nl v*' a + ...... + N /' a. Prove also the
result r,m p=Tn,m by contour integration.
Ex. ii. Prove that the function of f expressed by
has analogous properties to Riemann's normal elementary integral of the second kind.
Ex. iii. Prove that
where a, = («,a
358 FURTHER COMPARISON WITH A RIEMANN SURFACE.
Ex. iv. With the notation
prove that
[232
* (*, t«) - * (Z, f ) = 2*1 ^ /' " = *(*, £„) - * (Z, 5),
where £ is an arbitrary point, and hence prove that if z, clt ..., cp, £ be any arbitrary
points, and £1 = ^ (£), ..., £P = SP (£), the function of f expressed by
*(*, <T), *(*, I), *(*, &), .-., *(*, &)
(cp> 0, * (cp, '£), * (ep, £), . . . , * (cp, |p)
1 , 1 , 1 , .-., 1
is unchanged by the substitutions of the group, and has simple poles at z, c1? ... , cp, and
their analogues, and a simple zero at £, and its analogues. Thus the function is similar to
the function ty(x, a; 2, cl , ..., cp) of § 122, and every function which is unchanged by the
substitutions of the group can be expressed by means of it.
As a function of z, the function is infinite at z=£, z=£, beside being infinite at z=oo ,
and its analogues; when (aiz + pi)/(yiZ + 8i') is put for z, the function becomes multiplied
by (yi2 + $i)2- This last circumstance clearly corresponds with the fact (§ 123) that
\lr(x,a; z, clt ..., cp) is not a rational function of z, but a rational function multiplied by
Ex. v. Prove that
JEr. vi. In case p = l, we have
where
(a, - £)/(ar- J) = (Mei'c)r (a - J5)/(a - J ).
Putting, for abbreviation, q = el7rr = v pe1*, and
prove, by applying the fundamental transformation once, that
and shew that 0 (f) is a multiple of the Jacobian theta function 0 (/' a,q; £, £).
/-(D / ^"^?
jEr. vii. Taking two circles as in figure 6 (§ 223), let C'BfAC' = ir and I7i -r-^, = ^ ;
take an arbitrary real quantity a>, and a pure imaginary quantity a>' = ~ log p, and let
ITT
233] SOLUTION OF A PROBLEM OF HYDRODYNAMICS. 359
$>(u) denote Weierstrass's elliptic function of u with 2o>, 2<a' as periods. Then prove,
if a, c denote points outside both the circles, a' denote the inverse point of a in regard to
either one of the circles, and P, Q be arbitrary real quantities,
(a) that the function
* a-B lc-
is unaltered by the substitution (£' — B)/((' — A) = n(£-B)/(£—A), and has poles of the
first order, outside both the circles, only at the points f =a, f =c.
O) that the function,
__ P+iQ P-iQ
\ 1
r3 j *- 7?~| r3 T ^ 7?~! T^ 1 /• /J~| r* \ a' K~\
& 'r- log — |jp — • log - (f) -r- log - - . I — 0 I -^ \Q° - -7
[_Z7T (T £— A_] \_iir a a — A_j [_in & f — A_\ \_iir a ci'—A_\
is real on the circumference of each circle, and, outside both the circles, has a pole of the
first order only at the point f=«. The arbitraries P, Q can be used to prescribe the
residue at this pole.
Ex. viii. Prove that any two uniform functions of f having no discontinuities except
poles, which are unaltered by the substitutions of the group, are connected by an algebraic
relation (cf. § 235) ; and that, if these two be properly chosen, any other uniform function
of £ having no discontinuities except poles, which is unaltered by the substitutions of the
group, can be expressed rationally in terms of them. The development of the theory on
these lines is identical with the theory of rational functions on a Eiemann surface, but
is simpler on account of the absence of branch places. Thus for instance we have a
theory of fundamental integral functions, an integral function being one which is only
infinite in the poles of an arbitrarily chosen function x. And we can form a function such
as E (x, z} (§ 124, Chap. VII.) ; but the essential part of that function is much more
simply provided by the function, w (f, y), investigated in the following article.
233. The preceding investigations are sufficient to explain the analogy
between the present theory and that of a Riemann surface. We come now
to the result which is the main purpose of this chapter. In the equation
where {£, y/Zi, Ci] denotes a cross ratio, let the point z approach indefinitely
near to £ and the point c approach indefinitely near to 7; then separating
away the term belonging to the identical substitution, and associating with
the term belonging to any other substitution that belonging to the inverse
substitution, we have, after applying a linear transformation to every element
of the cross ratio arising from the inverse substitution
n*. < = ion- <* - P <c ~ ?> + vloo. (* - 0 (c< - 7) (* -Jt) (c - 7.-)
fcr --' 0---
where S' denotes that, in the summation, of terms arising by a substitution
360 INTRODUCTION OF THE FUNDAMENTAL FUNCTION [233
and its inverse, only one is to be taken, and the identical substitution is
excluded. Thus we have*
s=f,e=y
where IT has a similar signification to S' and {£, 7/7^, £*} denotes a cross
i i
ratio. Consider now the expression
«r(&7), = (C- 7) IT {£ 7/71. 6} 5
t
it has clearly the following properties — it represents a perfectly definite
function of £ and 7, single-valued on the whole £-plane ; it depends only on
two variables, and vr (£, 7) = — '57(7, £) ; as a function of £ it is infinite, save
for the singular points of the group, only at £=00, and not at the analogues
of £= oo ; it vanishes only at £ = 7 and the analogues of this point, and
limitf=y ts (£, 7)/(£— 7) = 1. Thus the function may be expected to generalise
the irreducible factor of the form x — a, in the case of rational functions, and
the factor <r (u — a) in the case of elliptic functions, and to serve as a prime
function for the functions of £ now under consideration (cf. also Chap. VII.
§ 129 and Chaps. XIII. and XIV.). It should be noticed that the value of
VT (£, 7) does not depend upon the choice we make in the product between
any substitution and its inverse ; this follows by applying the substitution
S r1 to every element of any factor.
234. We enquire now as to the behaviour of the function CT (£, 7) under
the substitutions of the group. It will be proved that
«!?», 7) +h g-^Y+K..)
,
where (— l^, (— !)*» are certain + signs to be explained.
This result can be obtained, save for a sign, from the definition of CT (£, 7),
as a limit, from the function II z] \ ; but since, for our purpose, it is essential
to avoid any such ambiguity, and because we wish to regard the function
CT (£, 7) as fundamental, we adopt the longer method of dealing directly with
the product (f— 7) 11' {£, 7/7$, &}. We imagine the barriers, each connecting
i
a pair of circles, which are necessary to render the functions v\a, ..., wp'a
* This function occurs in Schottky, Crelle, ci. (1887), p. 242 (at the top of the page). See
also p. 253, at the top. The function is modified, for a Eiemann surface, by Klein, Math. Annul.
xxxvi. (1890), p. 13. The modified function occurs also, in particular cases, in a paper by
Pick, Math. Annal. xxix., and in Klein, Math. Annal. xxxn. (1888), p. 367. For p = l, the
theta function was of course expressed in factors by Jacobi. The function employed by Bitter,
Math. Annal. XT,IV. (p. 291), has a somewhat different character.
234] OF THE THEORY OF THIS CHAPTER. 361
uniform, to be drawn; then the quantities rn>m, rn,n given in § 231, and
defined by vnm> , vnn- are definite; so therefore is also evlvny and the quan
tity ewiTn'n, which is equal to
IT
A n —
where s denotes a substitution, other than the identical substitution, not
beginning or ending with a power of *bn, and excluding the inverse of a
substitution which has already occurred. This formula raises the question
whether Kn, which we take positive, is to be regarded as less than 2?r or not,
since otherwise the sign of e*l'"» is not definite. But in fact, as it arises in
this formula, from vfa' f, log /*n + iKn is the value of log [ |j -£ I \, --5M when
\S ~ •»»/ t,—-"-nl
f ' has reached £n from f by a path which does not cross the barriers. Thus tcn
is perfectly definite when the barriers are drawn, and the sign of the
quantity
is perfectly definite and independent of the barriers. We denote it by
(- l)^"1. The annexed figure illustrates two ways of drawing a barrier
PP. In the first case /cn is less than 2?r. In the second case £' must pass
once round the point B, and KH is greater than 2?r. When Kn is thus
determined, the expression by means of Kn of the p£ which occurs in
the formulae connecting an> /3n, yn, 8n and An, Bn, pn, for instance in the
formula pn = (I + pn) / (an + Sn), is also definite; it may be /4=/4»eii<tn or
Pn = -pn e**n- We shall put p* = (- 1)^ g* ei^n If the whole investigation
had been commenced with a different sign for each of a.n, /?„, yn, 8n, hn would
have become hn — l, but gn, depending only on the circles and the barrier,
would have the same value.
We have
CT (£n, 7) = ?n - 7 n/ (Tin ~ 7 7»" ~ ?n ?j- ^
''
362 BEHAVIOUR OF THIS FUNCTION [234
where i denotes in turn all substitutions which with their inverses give the
whole group, except the identical substitution ; thus i denotes all substitutions
?ix for \= 1, 2, 3, ..., oo, as well as all substitutions nhsnk, where s has the
significance just explained and h, k take all positive and negative integer
values including zero. Therefore
- 7 7«* - £M £>- £
jj n*«i*_+ 1_^ 7
A, *, A £n>«im* — 7
£n - 7 jj £nA+l -7 jj £IA-£ jj 7nA ~ £»
(T- 7 A £«A - 7 A £»A+I - f A 7«x - f '
- (^n)n^— 7 (^.71X^8 — £
^, * (^H)HA* — 7 (Bn)nhs — £ h> s< k ynhmk — £ ^nhsnk+l — £n '
the transformation of the second part of the product being precisely as in the
first part,
- 7 Tn - 7 n~ A 7 - £«-
7 ^MMftg — „ 7 — ^M-ta-i
h, s nnhs — 7 (n)nhs ~ h,s,k 7 ~ £n-*sn-ft n ~
7?n-_g' _7_~(T (Tn ~ ^n (^n)n»» ~ 7 (.^!^!
- 7 7 ~
since & and — k have the same range of signification we may replace — k by h,
in the last form, and obtain, by a rearrangement of the second product,
., _
7) -5« - ?' 7 - ^n A, s ~ 7 - n
7 ~ (^n)
*, ^ 7
but, from the formula
^r=y, Z 7_-»)
» 27r. g-'~-'
where j can have the forms ?IAS, nhs-1, or be the identical substitution,
we have
**£ y= {-Bn 7 -^l, n C-(JgnU y-(An)nhg n ^-(^U-! 7-(^n))tA8-1 1
{—A* J - Bn h,^-(An)nhs ' 7 - (Bn)nhg s, h ^-(A^nhs-i ' 7 - (Bn)nk,-i '
235] UNDER A LINEAR SUBSTITUTION OF THE GROUP.
therefore
363
7)
— An s,h%n — (Bn)nha-\ ' £— (An)nh
-"-n
TT
S -"n s, &
*n ~ n
TT
n-ft — An
?t)s ~ An
and hence
(Bn)s-Bn-(An\-An'
CT
now from the formula (^ - 5B)/(C» - An) = Pn (S - Bn)/(?- An), and the
values of an> 0n, 7n, 8n given in § 226, we immediately find
or- ^»
thus, as pj = (- 1)^,4 e**"B, we have
hence, finally
7-n J
where (— \)°*e mrn.ne--lKn js independent of how the barriers are drawn, and
(-1)A"7«. (~ lA^i are independent of the signs attached to yn and Sn.
235. The function w(f, 7), whose properties have thus been deduced
immediately from its expression as an infinite product, supposed to be
convergent, may be regarded as fundamental. Thus, as can be imme
diately verified, the integral II£y is expressible by -07(^,7), in the form
y ,
»*(**)*<& tf)'
and thence the integrals v^y arise, by the definition v^y =
thence, also, integrals with algebraic infinities, by the definition
, and
~dxLi*'a
(cf. Ex. ii, § 232). Further, if F (£) denote any uniform function of % whose
value is unaltered by the substitutions of the group, which has no discontinui
ties except poles, it is easy to prove, by contour integration, as in the case of
364 EXAMPLES OF THIS FUNCTION. [235
a Riemann surface, (i) That F(£) must be somewhere infinite in the region S,
(ii) That F(£) takes any assigned value as many times within S as the sum
of its orders of infinity within 8, (iii) That if a1} ..., ak be the poles and
&, ..., fa the zeros of F (£) within S, and the barriers be supposed drawn,
where m1} ..., mp, m/, ..., mp' are definite integers. Thence it is easy to
shew that the ratio
is a constant for all values of £. And replacing some of /3lf ..., ctk in this
expression by suitable analogues, the exponential factor may be absorbed.
Ex. In the elliptic case where there is one fundamental substitution (£ — B)I(£' — A) =
p (f - B)I(£-A\ we have (& - 5)/(Cf - -4) = pi (f- ^)/(f- -4), and thence putting ?«, v, respec
tively for the integrals /, ^, so that e27rj'"= (f- ff)/((-A), cl™=(y- B)l(y- A], we
immediately find
~ Ct __ ! ~ 2Pi cos 2n" (M ~ y) +P2i >_. _
— - *2
7 ~~ y» y~ d (i ~ p*)2 2i s^n ""^ sin ""^
and hence
^?-^l sii
OT (C' y) ~ 2i sin TT« sin ^ ,2 (1 - p*)2
which*, putting enlT=p^, is equal to
where w is an arbitrary quantity, and
236. The further development of the theory of functions in the £ plane
may be carried out on the lines already followed in the case of the Riemann
surface. We limit ourselves to some indications in regard to matters bearing
on the main object of this chapter.
The excess of the number of zeros over the number of poles, in any
region, of a function of £, /(£), which is uniform and without essential
singularities within that region, is of course equal to the integral
See, for instance, Halphen, Fonct. Ellipt. (Paris, 1886), vol. i. p. 400.
230] INTRODUCTION OF THE THETA FUNCTIONS. 365
taken round the boundary of the region. If we consider, for example, the
function £ln(£), = dvn y/d£, which is nowhere infinite, in the region S, the
number of its zeros within the region 8 is
-'^n^pi
,(£r) a»(pja
where the dash denotes a differentiation in regard to f, and the sign of
summation means that the integral is taken round the circles C\' , ..., Cpf, in
a counter-clockwise direction. Since fln (fr) = (7r£ + 8r)z fln (£), the value is
or 2p; thus as Hn (f) vanishes to the second order at f= GO in virtue of the
denominator d£ we may say that <foj y has 2p - 2 zeros in the region £, in
general distinct from £ = x . The function flB (f) vanishes in every analogue
of these 2p - 2 places, but does not vanish in the analogues of £= co .
The theory of the theta functions, constructed from the integrals Vn y, and
their periods rn>m, will subsist, and, as in the case of the Riemann surface
there will, corresponding to an arbitrary point m, which we take in the
region S, be points ml} ..., mp in the region S, such that the zeros of the
function © (i£ m - «?• . ». - ...... - ^,™,.) are the places ^ ^ ^ ^ Anf}
corresponding to any odd half period, £OS),,, there will be places nli...inp_l,
in the region S, which, repeated, constitute the zero of a differential dtf> *, and
satisfy the equations typified by
The values of the quantities e«T»,« and the positions of m1} ...,mp may
vary when the barriers which are necessary to define the periods rn>m are
changed.
But it is one of the main results of the representation now under
consideration that a particular theta function is derivable immediately from
the function *r (f, 7) ; and hence, as is shewn in chapter XIV., that
any theta function can be so derived. Let v denote the integral whose
differential vanishes to the second order in each of the places n1} ....Wp.,.
Consider the expression */dv/d£ in the region S. It has no infinities and it is
single-valued in the neighbourhood of its zeros, as follows from the fact that
the p zeros of dv/d£ are all of the second order. Hence if the region S be
made simply connected by drawing the p barriers, and joining the p pairs of
circles byp-l further barriers (Cl), ... , (Cp_,), of which (cr) joins the circumfer
ence Cr' to the circumference Cr+l, ^/dvjd^ will be uniform in the region 8 so
long as £ does not cross any of the barriers. For the change in the value of
when f is taken round any closed circuit may then be obtained by
366 EXPRESSION OF w (£, 7) BY THETA FUNCTIONS. [236
considering the equivalent circuits enclosing the zeros. But in fact the
barriers (d), ..., (c^) are unnecessary; to see this it is sufficient to see that
any circuit in the region 8 which entirely surrounds a pair of circles, such
as GI, Clt encloses an even number of the infinities of dv/d£ which are at the
singular points of the group. Since these infinities are among the logarithmic
zeros and poles of v{' v, ..., vpy, whereof v is a linear function, the proof
required is included in the proof that any one of the functions vf Y, ..., v%y is
unaltered when taken round a circuit entirely surrounding a pair of the
circles, such as d', Q. Thus when the barriers which render the functions
V\ , ...,vf uniform are drawn, the function *Jdv/d£ is entirely definite within
the region S, save for an arbitrary constant multiplier, provided the sign of
the function be given for some one point in the region S. And, this being
done, if 7 be any point, the function y^y / is independent of this sign.
This function, with a certain constant multiplier, which will be afterwards
assigned, may be denoted by -^ (£).
237. We proceed now to prove the equation
;::
where s'v ' y = sfa ' y + ...... + sp'vf} y, and A is constant, independent of f and
7. It is clear first of all that the two sides of this equation have the same
poles and zeros in the region S. For ® (v*'y + £fls,S') vanishes to the first
order at the places 7, nlt ..., np_l} and ^(£) vanishes to the first order at
n1} ..., np_1} oo, while m (£, 7) vanishes to the first order at £=7, and is
infinite to the first order at £= QO *. Thus the quotient of the two sides of the
equation has no infinities within the region S. Further the square of this
quotient is uniform within the region S, independently of the barriers; for
this statement holds of each of the factors
«r(C,7), ^(0, 6(^ + ^11.,,), <Pr*^\
And, if f be replaced by £n, the square of the quotient of the two sides of the
equation becomes (cf. § 175, Chap. X.) multiplied by the factor
' 7»? + «» '
which is equal to unity. Nowf a function of £ which is unaltered by the
substitutions of the group, and is uniform within the region 8, and has no
* At the analogues of f = oo neither w (f, 7) nor I/ \j/ (f ) becomes infinite.
t If U+iV be the function, the integral \UdV, taken round the 2p fundamental circles is
expressible as a surface integral over S whose elements are positive or zero. In the case
considered the former integral vanishes.
238] INTRODUCTION OF AN ASSOCIATED FUNCTION X (f, 7). 367
infinities, must, like a rational function on a Riemann surface, bo a constant.
Since the square root of a constant is also a constant the proof of the equation
is complete.
From it we infer (i) that
* (£»)/* (0 - (- 1>""+A'1 (7H?+ «») (- l)*»,
and (ii) that the values of \fr (£) on the two sides of a barrier have a quotient
of the form (-!/». The constant factor to be attached to i/r(£) may be
chosen so that .4 = 1. P'or this it is sufficient to take for the integral v the
expression
where ®i (u) = d® (u)/dUi. Then (cf. § 188, p. 281) the right-hand side,
when f is near to 7, is equal to A (£- 7) + ..., while the left-hand side has
the value (£ — 7) -f ____
238. The developments of an equation analogous to that just obtained,
which will be given in Chap. XIV. in connection with the functions there
discussed, render it unnecessary for us to pursue the matter further here.
The following forms an interesting example of theta functions, of another kind.
Suppose that the quantities ^, . . . , pp are small enough to ensure (cf. § 226)
the convergence of the series
wherein p, denotes an arbitrary place within the region S, and i denotes a
summation extending to every substitution of the group. It will appear that
this function is definite in all cases in which the function «r (£, /*) is definite.
The function is immediately seen to verify the equations
and
where r denotes the substitution inverse to that denoted by i. Thus
X (£ M) = - X fa, f ).
The function has one pole in the region S, namely at p, and no other
infinities, and if the series be uniformly convergent near f = oo , as we assume,
368 PROPERTIES OF THE FUNCTION X (£ 7). [238
the function vanishes to the first order at £ = oc . The excess of the number
of its zeros over the number of its poles in 8, which is given by
i » ftx' (£»,/*) *•'(£,
where the dash denotes a differentiation in regard to f , and the integrals are
taken counter-clockwise round the circles C\r , ..., Cp', namely by
— . 2 I
is equal to p. Thus the function has p zeros in $ other than £ = oo ; denote
these by fa, ..., JJ,P. Within any region S-n$ the function has the analogue of
fj, for a pole, and the analogues of fa, ..., JJ,P for zeros; it does not vanish at
the analogue of £ = oc . This result may be verified also by investigating
similarly the excess of the number of zeros over the number of poles in any
such region ; the result is found to be p — 1.
Consider the ratio
where v is any linear function of th , ...,.tiu ; let £\, ..., ^2p_2 denote the
zeros of dv. Then /(£) is uniform within the region 8, and is unaltered by
the substitutions of the group. It has poles fj?, £, ..., £2p-2, and no other
infinities in S, and has zeros fa2, ..., fip2, the square of a symbol being written
to denote a zero or pole of the second order. Thus we have, precisely as for
the case of rational functions on a Riemann surface,
or (§179, p. 256),
(p2, £i, •••> (V-a)= (y"a2> •••, /V)>
and therefore, if mlt ..., mp denote the points in S, derivable from //, (§ 236),
such that ® (/' M - /" '"' - ...... - /" ' mp) vanishes in f = x, , . . . , £ = xp, we
have (§ 182, p. 265).
(fa2, ..., ftp*) = (mi2, ..., mp2).
When the barriers are drawn, let
v»"'"'+ ...... •\-vymp = ^(ki + klfT2i ! + ...... + kpTi>p), (i=l, 2, ...,p),
klt ..., kp, ki, ..., kp being integers.
Now consider the product X(£, jj,) •& (£, p,}. It has no poles, in 8, and its
zeros are fa, ..., fj,p. It is an uniform function of £ and, subjected to one of
the fundamental substitutions of the group it takes the factor
238] EXPRESSION BY THETA FUNCTIONS. 369
Hence the function
wherein &'/'M denotes Ar/t^H- ...... + V4'* and n denotes the j9 quantities
&i + &/T";, j + ...... +kpTiip, has, within S, no zeros or poles, and is such that,
for a fundamental substitution,
(cf. § 175, Chap. X.); thus, as in the previous article, F(%) is a constant
thus, also, gn + hn — kn is an even integer, = 2Hn, say, and we have
where P denotes the p quantities # + ^ + k^Ti> , + ...... + kp'ri>pt and A is
independent of $ But, if f describe the circumference Cn, the left-hand side
is unchanged, and the right-hand side obtains the factor «-"•'*'» Thus the
integers &/, ...,kp are all even ; put kr' = 2Hr'; then, as
where the notation is that of § 175, Chap. X., we have
wherein 5 is independent of £ and therefore, since the interchange of £ /*
leaves both sides unaltered, B is also independent of /j,. The value of 5 may
be expressed by putting £=/*; thence we obtain, finally,
This equation may be regarded as equivalent to 2^ equations. For if in
one of the p fundamental substitutions $>£=(«,-£ + &)/(7r£+ S,), we consider
the signs of o^, &., 7r, gr all reversed, the function X(£ /A), which involves the
first powers of these quantities, will take a different value. The function
•a (f, /*), the p fundamental circles, and the integrals /' M and their periods
rn,m, and therefore the integers g1} ..., gp, will remain unchanged, if the
barriers remain unaltered. But the integer hr will be increased by unity.
If, on the other hand, the coefficients a, /3, 7, B remaining unaltered,
one of the barriers be drawn differently, the left-hand side of the equation
remains unaltered; on the right-hand one of hlt ..., hp will be increased by
an integer, say, for example, hr increased by unity, and therefore each of
Ti.r, •••> fp,r also increased by unity. Putting u for v^^-^g-^h, and
B- 24
370 APPLICATION TO THE ELLIPTIC CASE. [238
neglecting integral increments of w, the exponent of the general term of the
theta series is increased, save for integral multiples of ZTTI, by
27Ti (— ^) nr + t7rar2,
which is an even multiple of iri, so that the general term is unchanged.
Ex. i. Prove that the function X (f, p.) can be written in the form
where the sign of summation refers to all the substitutions of the group, other than
the identical substitution, with the condition that when any substitution occurs its inverse
must not occur, and {f , & \ p, /z,-} denotes — — / * .
o r^t ' ot f^l
Ex. ii. In case /> = !, where the fundamental substitution is
(£'- B)/(£-A) = p(£-B)/(£-A),
putting e2™* = (f - B) / (f - A ), e2™ = (fJ.-B)/(fji-A), prove that
4 ( - l)Ai
(f, M) -
and hence
When A = 0 this becomes*
4i'&) sin TTM sin irv <r3 [2<o (M - y)]
o- [2o> (n - »)] '
f) sin2 TT (u-v}~\
(it-^) + p2< J'
where the sigma functions are formed with 2w, 2ar as periods, o> being an arbitrary
quantity. Thus (§ 235, Ex.)
where the symbol 50 is as in Halphen, Fonct. Ellip. (Paris, 1886), Vol. I. pp. 260, 252.
This agrees with the general result ; in piitting p^=eir'T we have taken g — \ ; and, as
stated, h is here taken zero.
When h = 1 we similarly find
... . 4i'o) sin iru sin irv <r3 [2o> (u — v + £)] - 2i)w (?t - ti)
*(»> A1)111 ' /~T> j\ ?~r T^O zr; ^\i~ e »
(Jj — A) ircr3 (<t>) cr \z,a) (u — V) J
and hence
also in agreement with the general formula. In these formulae Q(u] denotes the series
2e2iri«n+tirT»» = 1 + 2? COS (2JTM) -f- 2 J4 COS (47Ttt) + 2j9 COS (67TM) + ,
where q = el7TT.
* Cf. Halphen, Fonct. Ellip. (Paris, 1886), Vol. i. p. 422.
238] THE PROBLEM OF THE CONFORMAL REPRESENTATION. 371
Ex. iii. Denoting
where the summations include all substitutions of the group except the identical sub
stitution, respectively by um>n, v^n, prove that, when f is near to /*,
Ex. iv. If z, s be two single-valued functions of £, without essential singularities,
which are unaltered by the substitutions of the group, the algebraic * relation connecting
z and s may be associated with a Eiemann surface, whereon £ is an infinitely valued
function ; and if 2, s be properly chosen, any single-valued function of f without essential
singularities, which is unaltered by the substitutions of the group, is a rational function on
the Riemann surface. But if
where £'= -~ , etc., we immediately find that the value Z=(af + #)/(yf +8) gives
{Z, *} = {£, 2};
//cfe\2
therefore, as {£, 2}, = - {z, £} I ( ^- I , is a single- valued function of f without essential
/ V*i/
singularities, and is unaltered by the substitutions of the group, we have
{<T, 4 = 27(2, s),
where / denotes a rational function. Therefore, if Y denote an arbitrary function, and
P— - j^ log ( F2 ^ J , Y and £Y are the solutions of the equation
and if Ir be chosen so that F2 /^ is a rational function on the Riemann surface, the
coefficients in this equation will also be rational functions. Thus for instance we may
take for Y the function /y ^, in which case P=0, or we may take for Y the function
^ (0> = \/ j* j- » considered in § 236, which is uniform on the f plane when the barriers
are drawn, in which case P= - -^ log ~ , and the equation takes the form ^f + R.Y=0,
where R is a rational function, or again we may take for Y the uniform function of
C> * (f> /*). considered in § 238 f.
* Ex. viii. § 232.
t Cf. Riemann, Get. Werke (Leipzig, 1876), p. 416, p. 415; Schottky, Crelle, LXXXIII. (1877)
p. 336 ff.
24—2
372 THE HYPERELLIPTIC CASE. [238
Ex. v. If, as in Ex. iv., we suppose a Riemann surface constructed such that to
every point £ of the f plane there corresponds a place (z, *) of the Riemann surface, and
in particular to the point f =£ there corresponds the place (x, y\ and if R, S be functions
of £ denned by the expansions
prove that
and that R, 8 are rational functions of x and y.
Ex. vi. The last two examples suggest a problem of capital importance — given any
Riemann surface, to find a function f, which will effect a conformal representation of the
surface to such a f-region as that here discussed. This problem may be regarded as that
of finding a suitable form for the rational function I (z, a). The reader may consult
Schottky, Crelle, LXXXIII. (1877), p. 336, and Crelle, ci. (1887), p. 268, and Poincare',
A eta Mathem.atica, iv. (1884), p. 224, and Bulletin de la Soc. Math, de France, t. XI. (18 May,
1883), p. 112. In the elliptic case, taking
where $> denotes Weierstrass's function with 1 and T as periods, it is easy to prove that
-jr. and f */ -jj. are the solutions of the equation
239. There is one case of the theory which may be referred to in
conclusion. Take p circles Cf1, ..., Cp, exterior to one another, which are all
cut at right angles by another circle 0 ; take a further circle G cutting this
orthogonal circle 0 at right angles; invert the circles C1} C.2} ... in regard to
C. We shall obtain p circles <7/, C2', ..., Cp' also cutting the orthogonal
circle 0 at right angles. The case referred to is that in which the circles
C1} O/, ..., Cp, Cp are the fundamental circles and the angles Klt ..., KP are
all zero, so that, if ^n denote one of the p fundamental substitutions, the
corresponding points £, S-n£ lie on a circle through An and Bn. We may
suppose that the circles (7a , . . . , Cp are all interior to the circle C. It can be
shewn by elementary geometry that An, Bn are inverse points in regard to
the circle C as well as in regard to the circle Cn, and further that if &> denote
the process of inversion in regard to the circle C and wn that of inversion in
regard to Cn, the fundamental substitution S-n is a>nw, so that ufen<a = ST» , or
&)^7V = ^1&). Hence if the points of intersection of the circles 0, Cn be
called an', bn', the points of intersection of 0, Cn' be called an, bn, and the
points of intersection of 0, C be called a, b, it may be shewn without much
difficulty that
_ _
vr =
a,, . b.. , .^ a, b , T-, , -, c\ i \
nn' " = $ + Qn, vn' =$ + R, (n, r = l,2,...,p ; n+r),
239] THE HYPERELLIPfIC CASE. 373
where Pn> r, Qn, R are integers, and the integrations are along the perimeters
of the several circles. Hence it follows that the uniform functions of £
2jj£ c 2n^' c
expressed by e «••• *>r> e «- * are unaltered by the substitutions of the group.
Denote them, respectively, by xr (f) and x (£). Each of them has a single
pole of the second order, and a single zero of the second order, and therefore,
as in the case of rational functions on a hyperelliptic Riemann surface, we
have, absorbing a constant factor in xr (£), an equation of the form
x m-^O-^K)
•~*(0-*(&rj'
But it follows also that the function
is unaltered by the substitutions of the group. Hence we have*, writing
y, x for y (£), as (£), etc.,
if -xx
Thus the special case under consideration corresponds to a hyperelliptic
Riemann surface; and, for example, the equations t£"' 6"= 1 + Qn, etc., cor
respond to part of the results obtained in § 200, Chap. XI. It is manifest
that the theory is capable of great development. The reader may consult
Weber, Gottinger Nachrichten, 1886, "Ein Beitrag zu Poincare's Theorie,
u. s. w. ," also, Burnside, Proc. London Math. Soc. xxm. (1892), p. 283, and
Poincare, Acta Math. m. p. 80 and Acta Math. iv. p. 294 (1884); also
Schottky, Crelle, cvi. (1890), p. 199. For the general theory of automorphic
functions references are given by Forsyth, Theory of Functions (1893),
p. 619. The particular case considered in this chapter is intended only
to illustrate general ideas. From the point of view of the theory of this
volume, Chapter XIV. may be regarded as an introduction to the theory
of automorphic functions (cf. Klein, Math. Annalen, xxi. (1883), p. 141, and
Ritter, Math. Annalen, XLIV. (1894), p. 261).
* The function x here employed is not identical in case^ = l with the z of Ex. vi. § 238.
[240
CHAPTER XIII.
ON RADICAL FUNCTIONS.
240. THE reader is already familiar with the fact that if sn u represent
the ordinary Jacobian elliptic function, the square root of 1 — sn2 u may be
treated as a single-valued function of u. Such a property is possessed by
other square roots. Thus for instance we have*
v(l — sn u) (1 — k sn u)
l - 2om sin ~ + f* l - 2qm~* sin 4- g2"1-1
Tir - tjr \-rr ***
= M Sin -. (K - U) II
where M is a certain constant,' and, as usual, q = e~wR'IK. The single-
valuedness of the function V(l — sn u) (] — k sn u) can be immediately seen
to follow from the fact that each of the zeros and poles of the function
(1 — sn u) (1 — ksnu) is of the second order. It is manifest that we can
easily construct other functions having the same property. If now we write
u = ux> a and consider the square root on the dissected elliptic Riemann
surface, we shall thereby obtain a single-valued function of the place x,
whose values on the two sides of either period loop will have a ratio,
constant along that loop, which is equal to + 1.
Ex. Prove that the function
is a single-valued function of it.
Further we have, in Chapter XI., in dealing with the hyperelliptic case
associated with an equation of the form
y"- = (a; - Oj) . . . (x - a2p) (x - c),
* Cf. Cayley, Elliptic Functions (1876), Chap. XI. The function may be regarded as a
doubly periodic function, with 8K, 2iK' as its fundamental periods. It is of the fourth order,
with K, 5K, K + iK', 5K+iK' as zeros, and iK', 2K + IK', iK+iK', 6K+iK' as poles.
242] EXPRESSION BY RIEMANN INTEGRALS. 375
been led to the consideration of functions of the form V(c — x^) ... (c — xp),
which are expressible by theta functions with arguments u, =uxi>ai+ ......
+ U*P< aP. These functions are not only single- valued functions of the
arguments u, but, when the Riemann surface is dissected in the ordinary
way, also of every one of the places tK1} ... , scp. In fact the square root vc — x
is a single-valued function of the place x because, c being a branch place,
x — c vanishes to the second order at the place, and the point at infinity
being a branch place, x — c is there infinite to the second order. The values
of the square root \/c — x on the two sides of any period loop will have a
ratio, constant along that loop, which is equal to + 1.
241. More generally it may be proved, for any Riemann surface, that if
Z be a rational function such that each of its zeros and poles is of the mih
order, the mth root, \/Z, is a single-valued function of position on the
dissected surface, with factors at the period loops which are wth roots of
unity. And it is easy to prove this in another way by obtaining an ex
pression for such a function. For let alt ..., ar be the distinct poles of Z, and
fii, ..., @r its distinct zeros, so that the function is of order mr. Let Hz\ * be
the normal elementary integral of the third kind and Vi'a, ..., Vf the normal
integrals of the first kind. Then when the paths are restricted not to cross
the period loops we have* equations
kpfri>p,
wherein klt ..., kp, &/, . . . , kp' are certain integers independent of i. Hence
the expression
m [Hx> a A- 4- nx> a 1-2 'I- ' x> a - -2 -i- ' x> a
e 0u«l Pr> ar P P ,
wherein a is an arbitrary fixed place, represents the rational function Z, save
for an arbitrary constant ; and we have
,x,a
, ,.x, a
+ ...... +H ~—
'=Ae
where A is a certain constant. This expression defines \/Z on the dissected
surface as a single-valued function of position. More accurately it defines
one branch of \/Z, the other m- 1 branches being obtained by multiplying
A by wth roots of unity. So defined, the function V Z is affected, at the
-— K
period loop «,-, with a factor e m ', and, at the period loop «/, with the
factor em '.
242. We have, in chapters X., XL, been concerned with other functions,
namely the theta functions which also have the property of being single-
* Chap. VIII. § 155.
376 EXPRESSION BY THETA FUNCTIONS. [242
valued on the dissected Riemann surface, but affected with a factor for each
period loop. They are also simpler than rational functions, in that they do
not possess poles. It is therefore of interest to express such functions as
\/ Z by means of theta functions ; and the expression has an importance
arising from the fact that the theory of the theta functions may be established
independently of the theory of the algebraic integrals. To explain this
mode of representation consider the quotient
-/; 0
U1 . D\ '
- * ; ft)
where the numerator and denominator contain the same number of factors,
^ (a, q) denotes the function (Chap. X. § 189) given by
(n+g')2
q, r, ..., Q, R, ... denote any characteristics, and e,f....,E,F,... denote any
arguments.
Then by the formula (§ 190)
£ (u + flj, ; q) = &M <«>+2*W-J/'<?) ^ (M ; q)t
where M, M' denote integers, we have ty (u, + H „) fty (u) = e'\ where L is
\M(u-e) + \Jl(u-f) + ...... -\M(u-E)-\JI(u-F)- ......
+ 2iriM(q' + r'+ ...... - Q' - R' -...)- 2iriMf(q + r+ ...... -Q-R-...},
namely, is
...... -Q'-R'- ...)
q + r + ...... -Q-R-...).
Thus if
ei+fi+ ...... =Et + Fi+ ...... ,
and
ff* +rt + ...... -(Qi +Ri+...) = ~Ki, (i = l,2, ...,p),
I IV
qi+r^+ ...... -(Q/ + JR/+...) = -^/,
772*
where Kiy Kl are integers and m is an integer, it follows, for integral values
of M, M', that
If now we take b = iirr, as in § 192, and put ux<a for u, *$(u—e; q)
becomes a single- valued function of x whose zeros are (§§ 190 (L), 179) the
places #j, . .., xp, given by
244] COMPARISON WITH RATIONAL FUNCTIONS. 377
where alt ..., ap are p places determined from the place a, just as in § 179
the places m1 , ..., mp were determined from the place m ; hence, in this case,
i/r (w) is the wth root of a rational function, having for zeros places
each m times repeated, and for poles places
Y Y 7 7
.AH ..., -Ap, oif ..., &p, ...,
each m times repeated, these places being subject only to the conditions
expressed by the equations
A", (A).
II v
In this representation we have obtained a function of which the number
of m times repeated zeros is a multiple of p, and also the number of m times
repeated poles is a multiple of p. It is easy however to remove this restric
tion by supposing a certain number of the places x1} ..., xp, zlt ..., zp to
coincide with places of the set X1} ..., Xp, Z1} ..., Zp,
243. A rational function on the Riemann surface is characterised by the
facts that it is a single-valued function of position, such that itself and its
inverse have no infinities but poles, which has, moreover, the same value
at the two sides of any period loop. The functions we have described may
clearly be regarded as generalisations of the rational functions, the one new
property being that the values of the function at the two sides of any period
loop have a ratio, constant along that loop, which is a root of unity. For
these functions there holds a theorem, expressed by the equations (A) above,
which may be regarded as a generalisation of Abel's theorem for integrals
of the first kind ; and, when the poles of such a function are given, the
number of zeros that can be arbitrarily assigned is the same as for a rational
function having the same poles, being in general all but p of them; this
follows from the theory of the solution of Jacobi's inversion problem
(Chap. IX. ; cf. also §§ 37, 93). It will be seen in the course of the following
chapter that we can also consider functions of a still more general kind,
having constant factors at the period loops which are not roots of unity, and
possessing, beside poles, also essential singularities; such functions may be
called factorial functions. The particular functions so far considered may be
called radical functions ; it is proper to consider them first, in some detail, on
account of their geometrical interpretation and because they furnish a
convenient method of expressing the solution of several problems connected
with Jacobi's inversion problem.
244. The most important of the radical functions are those which are
square roots of rational functions, and in view of the general theory developed
in the next chapter it will be sufficient to confine ourselves to these functions.
s
378 ASSOCIATION OF A RADICAL FUNCTION [244
In dealing with these we shall adopt the invariant representation by means
of (^-polynomials, which has already been described*. An integral polynomial
of the rth degree in the p fundamental (^-polynomials, <f>1} ..., <j>p, will be
denoted by 3>{n , or "SP"'*, when its 2r(p — 1) zeros are subject to no condition.
When all the zeros are of the second order, and fall therefore, in general, at
r(p— 1) distinct places, the polynomial will be denoted by X(r> or Y(r) ; we
havef already been concerned with such polynomials, Xw , of the first degree
in </>!, ..., <j>p.
It is to be shewn now that the square root VJT(r) can properly be associated
with a certain characteristic of 2p half-integers; and for this purpose it is
convenient to utilise the places m1} ..., mp, arising from an arbitrary place m,
which have already | occurred in the theory of the theta functions. These
places are§ such that if a non-adjoint polynomial, A, of grade p, be taken to
vanish to the second order at m, there is an adjoint polynomial, ty, of grade
(11 — l)cr+ n — 3 + fi, vanishing in the remaining n/j, — 2 zeros of A, whose
other zeros consist of the places ml} ...,mp, each repeated. Take now any
(^-polynomial, <f>0> vanishing to the first order at m, and let its other zeros be
Al} ..., A^p-s", and take a polynomial <&(3) vanishing to the second order in
each of A1} ..., A2p_2; then <E>(3> will|| contain 5(^-1) - 2(2p - 3), = p+l,
linearly independent terms, and will have 6 (p — 1) — 2 (2p — 3), = 2p, further
zeros. Let X(l) be any ^-polynomial of which all the zeros are of the second
order. Consider the most general rational function, of order 2p, whose poles
consist of the place m, this being a pole of the second order, and of the zeros
of Xw. This function will contain 2p—p + 1, =p + 1, linearly independent
terms and can be expressed in either of the forms <!>(3'/</>02.X(1), -v|r/A^(1), where
•»Jr is any polynomial of grade (n — 1) cr + n — 3 + p which vanishes in the
up — 2 zeros of A other than m. Since now 11 -v/r can be chosen, = i|r, so that
the zeros of this function are the places m1} ...,mp, each repeated, it follows
that <E>(3) can be equally chosen so that this is the case. So chosen it may be
denoted by X(s>. Thus the places m^, ...,mp arise as the remaining zeros of a
form X{S} (with 3(p — 1), =p +2p — 3, zeros, each of the second order), whose
other 2p — 3 separate zeros are zeros of an arbitrary (^-polynomial, <)>0, which
vanishes once at the place m.
If now ?i:, ..., ?z_p_i be the places which, repeated, are the zeros of Xw, it
follows, since m, nl} ..., ?ip_i, each repeated, are the poles, and m1} ...,mp,
each repeated, are the zeros of a rational function, X{3}/<f>02Xw , that, upon the
dissected surface, we have
* Chap. VI. § 110 ff., and the references there given, and Klein, Math. Annal. xxxvi. p. 38.
t Chap. X. § 188, p. 281. I Chap. X. § 179.
§ Chap. X. § 183, Chap. VI. § 92, Ex. ix.
|| Chap. VI. § 111. H Chap. X. § 183.
245] WITH A HALF-INTEGER CHARACTERISTIC. 379
where ki, ..., kp, &/, ..., kp' are certain integers. Hence, as in § 241, it
immediately follows that the rational function Xi3)/(f)Q-X{11, save for a constant
factor, is the square of the function
and therefore that the expression ^X(3>/<f)0*/X(v may be regarded as a single-
valued function on the dissected Riemann surface, whose values on the two
sides of any period loop have a ratio constant along that loop. These constant
ratios are equal to enik>-' and e~n!kr for the rth loop of the first and second kind
respectively. When the places ml} ..., mp are regarded as given, these
equations associate with the form VZ (1) a definite characteristic
T «V> • • • j T"j» 9*1 t • • • > 2 "'P '
Also, if F|3> be any polynomial which, beside vanishing to the second
order in Alt ..., A^p_3, vanishes to the second order in places m/, ..., mp ,
Y(3]jX(3] is a rational function, and we have equations of the form
where \lt ..., \p are integers, A is a constant, and the paths of integration
are limited to the dissected Riemann surface. These equations associate
VF(3) with the characteristic \1} ..., \, X/, .,
And, as in § 184, Chap. X., we infer that every odd characteristic is
associated with a polynomial* Xw, and every even characteristic with a
polynomial F(3), which has Aly ..., A^^ for zeros of the second order; and it
may happen that the polynomial F(3) corresponding to an even characteristic
has the form </>02F(1), in which case the places w/, ..., mp consist of the place
m and the zeros of a form F(1) .
245. Let now X(*v+1) be any polynomial whose zeros consist of
(2i> + 1) (p — 1) places, zlt^, ..., each repeated ; let </>„ be as before, vanishing
in m, Alt ..., ^4.2p_3) and X(3} be as before, vanishing to the second order in
Al} ..., Azp-3, Wj, ..., wp. Then if 4>(I>) be any ^-polynomial whose zeros
are CL, C2) ..., the function
* Or in particular cases with a lot of such polynomials, giving rise to coresidual sets of
places.
380 GENERAL THEORY OF THE EXPRESSION [245
is a rational function of order 2 (2z/ -f 1) (p — 1) + 2, whose zeros are m, z1 , z.2, . . . ,
and whose poles consist of the places 7?^, ..., mp, and the zeros of <£'"), each
repeated. Hence as before <f>0*/X(Zv+v/<&w V5^> is a single-valued function on
the dissected surface, and the form \/X(Zv+l) is associated with a characteristic
^ql} ..., ^qp, \q{ , ..., ^qp , such that, on the dissected surface,
ZP,
(i = l, 2, .....p);
and if, instead of 4>(v), we had used any other polynomial ^{v>, the character
istic could, by Abel's theorem, only be affected by the addition of integers.
Suppose now that Y(^+l} is another polynomial, and take a polynomial
then if the characteristic of the function 0V7<^+1>^w\/X^ differ from that
of (/>0v X^+v !<&(") \/ X(z) only by integers, we have when xl} #2, • •• denote the
zeros of *Y^+l\ and d1} d2, ... denote the zeros of M^), the equation
xi i m\ Xp,m,t Xn+i.d, , , ,
P P '
MP'TPI
where Mlt ..., Mp> M^ , ..., Mp' denote integers; by adding this to the last
equation we infer* that ^^X^^T^^I^^ V(ti) X® is a rational function.
Hence "f1, since there exists a rational function of the form (f)^X^/X^3), we
infer, when *JX(2v+1\ ^Y^+1] have characteristics differing only by integers,
there exists a form <J><'X+|/+1> whose zeros are the separate zeros of \l X (2"+1) and
), and we have
Hence, all possible forms F'2^*1', with the same value of /i, whose
characteristics, save for integers, are the same, are expressible in the form
<j> (,*+„+!) /VZ(2"+1>, where <E><>*+*+1> is a polynomial of the degree indicated,
which vanishes once in the zeros of */X(2v+1}. All such forms -v/F^+D are
therefore expressible by such equations as
where V F](2'x+1), ..., v F"^^^ are special polynomials, and X,, ..., X2M(p_1) are
constants. The assignation of 2/j, (p — 1) — 1, = (2/i + 1) (p — 1) — p, zeros of
will determine the constants X1; ..., X^^-D, and therefore determine
the remaining p zeros. When p = 0 there may be a reduction in the number
of zeros determined by the others.
It follows also that the zeros of any form VF(2'A+1) are the remaining zeros
of a polynomial ^>^+2) which vanishes in the zeros of a form x/Z13' having
Chap. VIII. § 158. t Chap. VI. § 112.
246] OF RADICAL FUNCTIONS. 381
the same characteristic as \/F(2'1+1), or a characteristic differing from that of
integers. When the characteristic of ^/X® is odd, and
, we may take <J><"+2> to be of the form
It can be similarly shewn that if Xw be a polynomial of even degree, 2/j,,
in the fundamental 0-polynomials, of which all the zeros are of the second
order, and <f>w be any polynomial of degree /*, the quotient \/Xw/<&w may
be interpreted as a single-valued function on the dissected surface, and the
form vX(w may be associated with a certain characteristic of half-integers.
Further the zeros of ^Xw are the remaining zeros of a form c£<>*+1) which
vanishes in the zeros of a form VjT<-> of the same* characteristic as VZ" <*'*>.
Also if */Xm, \/F(1) be two forms whose (odd) characteristics have a sum
differing from the characteristic of VZ*2' by integers, the ratio
is a rational function ; and if we determine (p — 1) pairs of odd characteristics,
such that the sum of each pair is, save for integers, equal to the character
istic of \/Z<2', and V5V1', VF^, VZ^, VTy1"', ..., represent the corresponding
forms, there exists an equation of the form
As a matter of fact every characteristic, except the zero characteristic, can,
save for integers, be written as the sum of two odd characteristics in
2p_2 (2p-i_ 1) ways.
246. In illustration of these principles we consider briefly the geometrical
theory of a general plane quartic curve for which p = 3. We may suppose
the equation expressed homogeneously by the coordinates xl, x.2, x3 and take
the fundamental 0-polynomials to be cf>l = x1, (f>.2=x2, <f>s = cc3. There are
then 2^-1(2^-l) = 28 double tangents, X^, of fixed position. There are
2^, = 64, systems of cubic curves, X(3), each touching in six points. Of these
six points of contact of a cubic, X(3}, of prescribed characteristic, three may be
arbitrarily taken ; and we have in fact
= x, J^) + x, Vz/> + A, VIv5* +
where \lt A,,, A3, A4 are constants, and ^/X^, VJt2<3>, ..., are special forms of
the assigned characteristic. The points of contact of all cubics X(3} of given
odd characteristic are obtainable by drawing variable conies through the
points of contact of the double tangent, D, associated with that odd
characteristic. Let no be a certain one of these conies and let X0 denote the
corresponding contact-cubic ; then the rational function X0D/fl02 has, clearly,
no poles, and must be a constant, and therefore, absorbing the constant, we
infer that the equation of the fundamental quartic can be written
Or a characteristic differing from that of vjP by integers.
382 THE CASE OF THE BITANGENTS [246
Three of the conies through the points of contact of D are xj) = 0, x2D = 0,
x3D=Q; the corresponding forms of JT(3) are x^D, x.22D, xjD. Hence all
contact cubics of the same characteristic as VZ) are included in the formula
v z0,
or
where P = X,^ + \^cz + X3#3, X1; Xa, X3 being constants ; the conic through the
points of contact of D which passes through the points of contact of X® is
given by H = 2 \/J9X<3', or H = 2PD + H0 ; and the fundamental quartic can
equally be written
4,x®D - n2 = 4 (XQ + n0p + DP*) £ - (n0 + ZPD? = o.
If then we introduce space coordinates X, Y, Z, T given by
X=x1, Y=x Z=x T=
so that the general form of VJT(3) with the same characteristic as VZ) is given
by
VZ<3> = Vi) (X1Z + X27 + X3^- T),
we have
4Z0 (Z, F, Z) D (Z, F, £) = W (X, 7, Z),
2TD (X, Y, Z) + A, (X, Y, Z) = 0,
where X0 (X, Y, Z) is the result of substituting in X0, for aclt oc2, x3,
respectively X, Y, Z, etc. ; by these equations the fundamental quartic is
related to a curve of the sixth order in space of three dimensions, given
by the intersection of the quadric surface
and the quartic cone
4Z0 (X, Y, Z) D (X, F, Z) = Cl* (X, Y, Z) •
the curve lies also on the cubic surface
T*D (X, Y, Z) + mo (X, Y, Z) + X0 (X, Y, Z) = 0,
which can also be written
(T-P)*D(X, 7,Z) + (T-P)Q(X, Y,Z) + X®(X, Y, Z) = 0,
where P denotes \,X + X2F+ \3Z, n = 2PD + flc, and X® = DP2 + H0P + X0,
as above.
It can be immediately shewn (i) that the enveloping cone of the cubic
surface just obtained, whose vertex is the point X = 0 = F= Z, is the quartic
cone whose intersection with the plane T = 0 gives the fundamental quartic
curve, (ii) that the tangent plane of the cubic surface at the point
247] OF A PLANE QUARTIC CURVE. 383
X = 0 = Y = Z is the plane D (X, Y, Z) = 0, (iii) that the planes joining
the point X = 0 = F = Z to the 27 straight lines of the cubic surface
intersect the plane T=0 in the 27 double tangents of the fundamental
quartic other than Dt (iv) that the fundamental quartic curve may be
considered as arising by the intersection of an arbitrary plane with the
quartic cone of contact which can be drawn to an arbitrary cubic surface
from an arbitrary point of the surface.
Thus the theory of the bitangents is reducible to the theory of the right
lines lying on a cubic surface. Further development must be sought in geo
metrical treatises. Cf. Geiser, Math. Annal. Bd. I. p. 129, Crelle LXXII. (1870);
also Frahm, Math. Annal. vn. and Toeplitz, Math. Annal. XL; Salmon, Higher
Plane Curves (1879), p. 231, note ; Klein, Math. Annal. xxxvi. p. 51.
247. We have shewn that there are 28 double tangents each associated
with one of the odd characteristics ; the association depends upon the mode
of dissection of the fundamental Riemann surface. We have stated moreover
(§ 205, Chap. XL), in anticipation of a result which is to be proved later, that
there are 8 . 36 = 288 ways in which all possible characteristics can be repre
sented by combinations of one, two, or three of seven fundamental odd
characteristics. These fundamental characteristics can be denoted by the
numbers 1, 2, 3, 4, 5, 6, 7, and in what follows we shall, for the sake of
definiteness, suppose them to be either the characteristics so denoted in the
table given § 205, or one of the seven sets whose letter notation is given at
the conclusion of § 205. Thus the sum of these seven characteristics is the
characteristic, which, save for integers, has all its elements zero ; or, as we
may say, the sum of these characteristics is zero.
A double tangent whose characteristic is denoted by the number i will be
represented by the equation m = 0. A combination of two numbers also
represents an odd characteristic (§ 205, Chap. XL), so that there will also be
21 double tangents whose equations are of such forms as uitj = 0. The three
products Vi^ttj,, 4/t%»,7, *Ju3ul2 will be radical forms, such as have been denoted
by V3>, each with the characteristic 123. Hence if suitable numerical
multipliers be absorbed in ul} u3, we have (§ 245) an identity of the forms
vVw^ + Vi*2w31 + V-M.WU = 0, (u,u3l + u3u12 - ulU23)2 = 4,u2u3u3^2 ;
this must then be a form into which the equation of the fundamental quartic
curve can be put. Further, each of the six forms
has the same characteristic, denoted by the symbol 1. Thus, if suitable
numerical multipliers be absorbed in u2, u4, the equation of the quartic can
also be given in the form
384 THE CASE OF THE BITANGENTS [247
If therefore
/= UaUsi + U3U12 — WiW23> <j) - U2u12 + U^uu — U3U13,
we have
(/- 0) (/+ <#>) = 4u2u12 (wsM,, - w,4w14).
Now if/-<£ were divisible by u2, and/+</> divisible by ul2, the common
point of the tangents u2 = 0, u12 = 0 would make /= 0, and therefore be upon
the fundamental quartic, /2 = 4w2w3w31w12 ; this is impossible when the quartic
is perfectly general. Hence, without loss of generality, we may take
f—<f> =
2
X being a certain constant, and therefore
*u2u12, = u3uls - X (u.2u3l + usu12 - u-^
Therefore, when the six tangents uly u2, u3, u^, uzl, ulz are given, the tangents
ut, w]4 can be found by expressing the condition that the right-hand side
should be a product of linear factors ; as the right-hand is a quadric function
of the coordinates this will lead to a sextic equation in X, having the roots
X = 0, X = oo ; if the other roots be substituted in turn on the right-hand, we
shall obtain in turn four pairs of double tangents ; these are in fact (u4, Uu),
(us, u15), (ue, ULG), (UT, ul7). We use the equation obtained however in a
different way ; by a similar proof we clearly obtain the three equations
W4W14 = U3U13 — Xj (U2U31 + U3U12 — l^U^} + X^WgWjj,
u^u^ = UiUn - X2 (u3u12 + u^i^ - u2u31) + X22M3W23, (B)
\s (U^zs + U2U31 — U3U12) + X32^ U31,
and hence
from this we infer that the common point of the tangents wlT u4 either lies on
11
w23 or on X2u3 + -2 = Q; as the fundamental quartic may be written in the
A-3
form ^AU^UM + \/Bu2u^ + VC^t^ = 0, it follows that if ^, M4, u.23 intersect,
they intersect on the quartic, which is impossible. Hence w4 must pass
through the intersection of ^and X2w3 + -^ = 0 ; now we may assume that
X3
the tangents uly u2, u3 are not concurrent, since else, as follows from the
equation Vw^ + Vw2w31 + *Ju3u12 = 0, they would intersect upon the quartic ;
thus w4 may be expressed linearly by u1} u2, u3, and we may put
+ a2u2 + asu3 = a^ + -x^Us +~)
X3X
247] OF A PLANE QUARTIC CURVE. 385
and so obtain X2=Aia3, X3 = 1/^a.j, hi being a certain constant; then the
equation under consideration becomes
or
/M24 7/34 , \ /^,
U* V7u X~ ~ lW23J = Ml I XT + ^Wsi " /2
so that, if &! denote a proper constant,
, ^34 _ 7 &1
I -v - Wl "^S 7 Wl»
A.3 fli
ci3 &%
We can similarly obtain the equations
- k2u, = ^ + ^- h,u3i (2 + a
where /i2) /<3, Ar2, A?3 are proper constants; therefore, as u.a, u3l, ua are not
concurrent tangents, since else they would intersect on the fundamental
quartic, we infer, by comparing the right-hand sides in these three equations,
and hence, k1 = k.2 = ks, = k, say, and 1 + 2^ + a,2/*,2 = 0 or hi = - i
a,
7 1 I 1
//,, = -- , *,- -- .
tt2 a:i
Thus
-fa«l=2» + ^! + ^>
ttj a., as
or
w.,8 ««31 7^],
— +-+—" + ^ («! Wj + «2 u2 + a3u3) = 0. (C)
u-i a2 t'3
Further we obtained the equation
"24^34 A?j
_ + _ = /,1^__Wi;
thus we have
25
386 THE CASE OF THE BITANGENTS [247
and therefore, as Xj = -- -, X3 = -- -, and similarly Xj = — -, we have, by
ft] (12 d3
the equation (C),
-- -uu = — + k (a2u2 + a3u3),
ct/% (LI
Cvj ^31 7 / \
-- M24 = -- 1- fc (a3u3 + aiWj),
CL3 CL2
2 12 7 / \
^34 ^~ - I iG \ ^1 ^1 I ^2 ^2/'
tti CL3
But if we put
u5= b1u1 + b2u2 + b3it3, ^£e = c1'M1 + c2M2 + c3w3, U7=dlul+d2u2 + d3u3,
we have also three other equations such as (C), differing from (C) in the
substitution respectively of the coefficients b1} b.2) b3 clf c2, C3 and dlt d2, ds in
place of alt a2, a3, and of three constants, say I, m, n, in place of k. As the
tangents u5, u6, u7 are not concurrent (for the fundamental quartic can be
written in a form v usul5 + *Ju6uls + ^u7u17 = 0) we may use these three last
equations to determine u23, u3l, u12 in terms of w1} w2, us; the expressions
obtained must satisfy the equation (C). Thus there exist, with suitable
values of the multipliers A, B, C, D, the six equations
A B C D
Dndl = 0,
/i Ui ] C/i
A B C D
+ Dnd2 = 0,
ct% 02 c2 d2
A B C D
- + j- + -- + -j = 0, Aka3 + Blb3 + Cmc3 + Dnd3 = 0.
C13 03 C3 Cl3
From these equations the ratios of the constants k, I, m, n are determinable;
suppose the values obtained to be written pk', pi', pm', pn', where p is undeter
mined, and k', I', m, n' are definite ; then, if we put a; for af VK, & for
biVl', <yt for C;Vm', Si for di^ln, v^ for u^/p, vsl for ti3l/p, and v12 for u12/p, the
equations obtained consist of
(i) four of the form
Vos VS1 V,o //^(,,
-+~ + --+alul + a2u2 + a3u3 = 0 (C )
ai a2 a3
in which there occur in turn the sets of coefficients (alt «2, a3), (y81} ^2, /93),
(71. 72, 7s)> (^i, S2, S3) ; from any three of these v^, v3l, vl2 may be expressed in
terms of u1} u2, us;
(ii) four sets of the form
where v,4 = uujp ^k', v^ = u^/p VF, VM = u^/p VF.
247] OF A PLANE QUARTIC CURVE. 387
It will be recalled that in the course of the analysis the absolute values,
and not merely the ratios of the coefficients in u^, u2, u3, u4, u^, u6, u?, have
been definitely fixed. Thus when these seven bitangents are given the
values of alt a,, a3, blt b,, ba, etc. are definite ; therefore the equations of the
15 bitangents v.a, v3l, vn, vu> v», VM, ...... are now determined from the seven
given ones in an unique manner, and there is an unique quartic curve
expressed by
12 = 0,
which has the seven given lines as bitangents.
It remains now to determine the remaining six double tangents whose
characteristics are denoted by
45, 46, 47, 56, 57, 67.
If the characteristics 1, 2, 3, 4, 5, 6, 7 be taken in the order 1, 4, 5, 2, 3, 6, 7
it is clear that as we have determined the double tangents u^, u3l, ul2 in
terms of ul} u», u3, so we can determine the tangents u^, usl, uu in terms
of ul} ut, u5. Thus the tangent «« can be found by substitutions in the
foregoing work. For the actual deduction the reader is referred* to the
original memoir, Riemann, Ges. Werke (Leipzig, 1876), p. 471, or Weber,
Theorie der Abel'schen Functionen vom Geschlecht 3 (Berlin, 1876), pp. 98—100.
Putting ctlul = x, o2-M2 = y, a3u3 = z, i^fa = £, %/«, = ??, v12/a3=^, /3i/<xi = Ai,
yi/di = Bi, Bifai = Gi (i= 1, 2, 3), the quartic has the form
= 0,
and the 28 double tangents are given by the following scheme, where the
number representing the characteristic is prefixed to each
(1) *=0, (2) y = 0, (3) *=0, (23) | = 0, (31) 7; = 0, (12) £=0,
(4) x + y+z = 0, (5) A1x + A<,y+A3z = 0, (6) B^x + B^y + B3z= 0,
(7) (7^ + 0^ + 6^ = 0,
(14) % + y + z = 0, (24) ^+z + x = 0, (34) £ + x + y = 0,
(15) - + A,y + A3z = Q, (25) -£• + A3z + AlX=0, (35)
AI A-i --
(16) L + B.2y + B3z = 0, (26) £ + B3z + B1x=Q, (36) -^ + B& + B,y = 0,
Jji E*i **>
(17) | + C,y + G3z = 0, (27) ^- + G3z + C,x = 0, (37) £ + C,x + C,y = 0,
GI G2 ^3
* For the theory of the plane quartic curve reference may be made to geometrical treatises ;
developments in connection with the theta functions are given by Schottky, Crelle, cv. (1889),
Frobenius, Crelle, xcix. (1885) and ibid. cm. (1887) ; see also Cayley, Crelle, xciv. and Kohn,
Crelle, cvn. (1890), where references to the geometrical literature will be found.
25—2
388 THE CASE OF THE BITANGENTS [247
* *
(67)
(75)
- _ _
J- — ^In-Aft A — -/la^li J. —
___
(d,(i\
__
51 (1 - B,BZ} B, (1 -
1 - Ca C3 1 - (73 ^ 1 - 0,
(47}
__ __ __ __
, (1 - G2G3) C, (1 - ^00 C, (1 - C
Here the six quantities x, y, z, g, i), £ are connected by the equations
= 0,
rr + 7T + rr + CiX+C,y+C3z = 0.
U, l/j O3
Conversely, if we take arbitrary constants ^.,, ^2, A3, 5,, 52, £3, whose
number, 6, is, when ^ = 3, equal to Sp — 3, namely equal to the number
of absolute constants upon which a Riemann surface depends when p = 3,
and, by the first three of the equations (D) determine £, 77, f in terms of the
arbitrary lines x, y, z, the last of the equations (D) will determine Clt C.,, C3
save for a sign which is the same for all ; then it can be directly verified
algebraically that the 28 lines here given are double tangents of the quartic
curve V#£ + \/yt] + \/z%= 0.
248. Before leaving this matter we desire to point out further the
connection between the two representations of the tangents which have been
given. Comparing the two equations of the fundamental quartic curve
expressed by the equations (§§ 246, 247)
and putting, in accordance therewith,
D (xl , x2 , x3} = £ H0 (xl , x., , as,) = z£- x% - yq, X0 (xl , x2 , a?,) = xyq
and (cf. p. 382) replacing the fourth coordinate T by T + u, where
249] OF A PLANE QUARTIC CURVE. 389
u is an arbitrary linear function of x, y, z or xlt x2, x3) the equation of the
cubic surface
(T+ ™)2 D + (T+ u) 00 + X0 = 0,
becomes
T*£ + T (z%- x% - y-n + 2i*f ) + u*% + u (z% - y<n - x%} + xyrj = 0,
or
which will be found to be the same as
Write now
v = u — x — z, w = u — x — % , u = u — x, v' = u+y, iv' = u+ rj;
then we obtain the result, easy to verify, that if u, v, w, u', v', w' be arbitrary
linear functions of the homogeneous space coordinates X, Y, Z, and T be
the fourth coordinate, the tangent cone to the cubic surface*
(T + u)(T+v)(T+w)-(T+u')(T + J)(T+v/) = Q (i)
from the vertex X = 0 = Y= Z can be written in the form
V(P - P) (u- u'} + \/(u - v') (u - w') + V(M' - v) (u' - w) = 0,
where P — P' = u + v + w — u' — v' — w' ; we have in fact
x = u — u', y =v' — u, z = u' — v, 77 = w' — u, %=u' — w,
Now the 27 lines on the cubic surface (i) can be easily obtainedf; and
thence the forms obtained in § 247, for the bitangents of the quartic, can be
otherwise established.
249. Ex. i. Prove that when the sum of the characteristics of three bitangents of the
quartic is an even characteristic, their points of contact do not lie upon a conic.
By enumerating the constants we infer that it is possible to describe a plane quartic
curve having seven arbitrary lines as double tangents. By the investigation of § 247
it follows that only one such quartic can be described when the condition is introduced
that no three of the tangents shall have their points of contact upon a conic. By the
theory here developed it follows that for a given quartic such a set of seven bitangents can
be selected in 8 . 36 = 288 ways.
Ex. ii. We have given an expression for the general radical form \/A'(3> of any given
odd characteristic. Prove that a radical form \/XM whose characteristic is even, denoted,
suppose, by the index 123, can be written in the form
* Any cubic surface can be brought into this form, Salmon, Solid Geometry (1882), § 533.
t See Frost, Solid Geometry (188(5), § 537. The three last equations (D) of § 247 are deducible
from the equations occurring in Frost. The three equations correspond to the three roots of the
cubic equation used by Frost.
390 NOETHER'S GENERAL SOLUTION [249
where X, A1? X2, X3 are constants, and MJ, M^- denote double tangents of the characteristics
denoted by the suffixes, as in § 247.
Ex. iii. If (^q, £<?')> (ir? iO denote any two odd characteristics of half-integers,
express the quotient
algebraically, when p = 3.
Ex. iv. Obtain an expression of the quotient of any two radical forms \/ X$\
of assigned characteristics and known zeros, by means of theta functions,/* being equal to 3.
250. Noether has given* an expression for the solution of the inversion
problem in the general case in terms of radical forms, which is of importance
as being capable of great generalization.
Using the places m1} ..., mp, associated as in Chap. X. with an arbitrary
place m, and supposing them, each repeated, to be the remaining zeros of a
form X (3), which vanishes to the second order in each of the places Alt ..., A.2p_3
in which an arbitrary ^-polynomial, <£0, which vanishes in m, further vanishes,
as in § 244, let VF(3) be any radical form, and <I>(1) any (^-polynomial whose
zeros are a1} ... , 0^-2- Then (§ 241) the consideration of the rational function
</>02F<3)/[<E>(1)]2^(3) leads to the equations
wherein the places
X} , . . . , &-2p— 3 , C'i , . . . , Cp
are the zeros of \/F(3), all of o-u ..., crp, a/, ..., arp' are integers, and z is an
arbitrary place; and, as follows from these equations, the places xlt .•.,x2p-3
may be arbitrarily assigned, the places c1; ..., cp and the form \/F(3> being
determinate, respectively, from these equations and the equation
, 0 ^.. -f-.. rt' •n'
g5^W]P> =C 1>a' ...... '"*" -'+ '"'"' + ...... + e"'m
• r / •' ' ^ / -^j ^n
+ Tn[<rlvl + ...... + <rp vp J,
wherein the place a is arbitrary. Hence if we speak of
as the characteristic of VF(3), it follows, if \IZ(® be another radical form with
the characteristic
and the zeros
Xl, . . . , ^2p_3 , ttj , . . . , Clp ,
* Math. Annal. xxvm. (1887), p. 354, "Zum Umkehrproblem in der Theorie der Abel'schen
Functionen."
250] OF THE INVERSION PROBLEM
that the quotient */Y®/*/Z(3), which is equal to
391
wherein A is a quantity independent of x, is (§ 187, Chap. X.) also equal to
_: r/ _ / _ - /\ ~j%t ^4- -|- ((T ' — fi f\ v^' ^1 (—\ / %> m GI j "^1 <i/^* * "*P\
(7e d ~d m '
where (7 is a quantity independent of x ; but by the equations here given
this is the same as
. {l , M *, « , , / ; >\ „,#, a-,
vi[(ffl-pl)vl + + (*!, -P,,)vp' ]
Ce
where £ H^ denotes p such quantities as J(<7{ + CT/T,:, j + + o-p'ritp); thus,
if we put
and recall the formula (§ 175)
we infer that
where E is a quantity independent of x.
Now in fact (§ 245) the general radical form \/F(3), of assigned charac
teristic (^cr, ^<r')> is given by
/ (3) / (3)
where ^ F! ,...,*«' F2p_ 2 are special forms of this characteristic, and Xj , . . . , X2^2
are constants. If we introduce the condition that VF(3) vanishes at the
places x1, ...,#.,p_3 we infer that VF(3) is equal to F^ (x, xly ..., #2p_3), where
/Q\
F is independent of x and Aa (x, xlt ..., #21>_3) denotes the determinant
in which t is to be taken in turn equal to 1, 2, ..., 2p — 3. Hence we have
(3)
, _
~~
392 BY MEANS OF RADICAL FUNCTIONS. [250
where, from the symmetry in regard to the places x, xly ..., x.,p_3, G is
independent* of the position of any of these places, and v is given by
y = lf> «2p- 2-j-fl*!. «i + ...... _|_^2p-3. «2j>-3.
To apply this equation to the solution of the inversion problem expressed
by p such equations as
Vx> ' *> + ...... + Vxi" ^P — U,
where pl, ..., ^p denote p arbitrary given places, we suppose the positions of
the places xp+l, ..., x2p_3 to be given ; then instead of &„(%, xly ..., #2p_3) we
have an expression of the form
where v } 7[ (a), ..., v Yp+i(x) denote forms \/F(3) (x) vanishing in the given
places asp+1, ..., xw_z, and A1} ..., Ap+1 are unknown constants. Since the
arguments u are given, the arguments v are of the form if' a%>-2 + w, where w
is known. If then in the equation
we determine the unknown ratios A 1 : A2 : ...... : Ap+1 : B± : ...... : Bp+\
by the substitution of 2p + 1 different positions for the place x, this equation
itself will determine the places xlt ..., xp. They are, in fact, the zeros of
either of the forms
other than the given zeros xp+l) ..., ^2p_3. If the first of these forms be
multiplied by an arbitrary form \/F(3) (x), of characteristic (Jo-, ^cr'), the
places #a, ..., xp are given as the zeros of a rational function of the form
of which 4>p — 6 zeros are known, consisting, namely, of the places xp+1, ..., Xy,-*
and the zeros of ViT(3) (x).
In regard to this result the reader may consult Weber, Theorie der A bePschen Functio-
nen vom Geschlecht 3 (Berlin, 1876), p. 157, the paper of Noether (Math. Annal. xxvm.)
already referred to, and, for a solution in which the radical forms are with roots of rational
functions, Stahl, Crelle, LXXXIX. (1880), p. 179, and Crette, cxi. (1893), p. 104. It will be
seen in the following chapter that the results may be deduced from another result of
a simpler character (§ 274).
251. The theory of radical functions has far-reaching geometrical applications to
problems of the contact of curves. See, for instance, Clebsch, Crelle, LXIII. (1864), p. 189.
For the theory of the solution of the final algebraic equations see Clebsch and Gordan,
Abel'sche Functnen. (Leipzig, 1866), Chap. X. Die Theilung; Jordan, Traite des Sub
stitutions (Paris, 1870), p. 354, etc.; and now (Aug. 1896), for the bitangents in case p = 3,
Weber, Lehrbuch der Algebra (Braunschweig, 1896), II. p. 380.
* For the determination of G see Noether, Math. Annal. xxvm. (1887), p. 368, and Klein,
Math. Annal. xxxvi. (1890), pp. 73, 74.
252]
CHAPTER XIV.
FACTORIAL FUNCTIONS.
252. THE present chapter is concerned* with a generalisation of the
theory of rational functions and their integrals. As in that case, it is conve
nient to consider the integrals and the functions together from the first. In
order, therefore, that the reader may be better able to follow the course of
the argument, it is desirable to explain, briefly, at starting, the results
obtained. All the functions and integrals considered have certain fixed
singularities, at placesf denoted by clf ..., ck. A function or integral which
has no infinities except at these fixed singularities is described as everywhere
finite. The functions of this theory which replace the rational functions of
the simpler theory have, beside the fixed singularities, no infinities except
poles. But the functions differ from rational functions in that their values
are not the same at the two sides of any period loop ; these values have a
ratio, described as the factor, which i^ constant along the loop ; and a system
of functions is characterised by the values of its factors. We consider two
sets of factors, and, correspondingly, two sets of factorial functions, those of
the primary system and those of the associated system; their relations are
quite reciprocal. We have then a circumstance to which the theory of
rational functions offers no parallel ; there may be everywhere finite factorial
functions^.. The number of such functions of the primary system which are
linearly independent is denoted by cr' + 1 ; the number of the associated
system by a- + 1. As in the case of algebraical integrals, we may have every
where finite factorial integrals. The number of such integrals of the primary
system which are linearly independent is denoted by CT, that of the associated
system by CT'. The factorial integrals of the primary system are not integrals
of factorial functions of that system ; they are chosen so that the values u, u'
* The subject of the present chapter has been considered by Prym, Crelle, LXX. (1869), p. 354;
Appell, Acta Mathematica, xm. (1890); Hitter, Math. Annal. XLIV. (1894), pp. 261—374. In
these papers other references will be found. See also Hurwitz, Math. Annal. XLI. (1893), p. 434,
and, for a related theory, not considered in the present chapter, Hurwitz, Math. Annal. xxxix.
(1891), p. 1. For the latter part of the chapter see the references given in §§ 273, 274, 279.
t In particular the theory includes the case when & = 0, and no such places enter.
£ This statement is made in view of the comparison instituted between the development of
the theory of rational functions and that of factorial functions. The factorial functions have
(unless k = Q) fixed infinities.
394 SUMMARY OF RESULTS. [252
of such an integral on the two sides of a period loop are connected by an
equation of the form u' = Mu + p,, where p, is a constant and M is the factor of
the primary system of factorial functions which is associated with that period
loop. The primary and associated systems are so related that if F be a
factorial function, of either system, and G' a factorial integral of the other
system, FdG'jdx is a rational function without assigned singularities. In the
case of the rational functions, the smallest number of arbitrary assigned poles
for which a function can always be constructed is p + 1. In the present
theory, as has been said, it may be possible to construct factorial functions of
the primary system Avithout poles ; but when that is impossible, or <r' + 1 = 0,
the smallest number of arbitrary poles for which a factorial function of the
primary system can always be constructed is or' + 1. Similarly when
a- + 1 = 0, the smallest number of arbitrary poles for which a factorial func
tion of the associated system can always be constructed is OT + 1. Of the
two numbers cr + 1, cr'+l, at least one is always zero, except in one case,
when they are both unity. When o-' + l is > 0, the everywhere finite fac
torial functions of the primary system can be expressed linearly in terms of
the everywhere finite factorial integrals of the same system. We can also
construct factorial integrals of the primary system, which, beside the fixed
singularities, have assigned poles ; the least number of poles of arbitrary
position for which this can be done is a- + 2. And we can construct factorial
integrals of the primary system which have arbitrary logarithmic infinities ;
the least number of such infinities of arbitrary position is cr + 2. For the
associated system of factors the corresponding numbers are cr' + 2.
It will be found that all the formulae of the general theory are not imme
diately applicable to the ordinary theory of rational functions and their
integrals. The exceptions, and the reasons for them, are pointed out in
footnotes.
The deduction of these results occupies §§ 253 — 267 of this chapter. The
section of the chapter which occupies §§ 271 — 278, deals, by examples, with
the connection of the present theory with the theory of the Biemann theta
functions. With a more detailed theory of factorial functions this section
would be capable of very great development. The concluding section of the
chapter deals very briefly with the identification of the present theory with
the theory of automorphic functions.
253. Let G!, ..., Ck be arbitrary fixed places of the Riemann surface,
which we suppose to be finite places and not branch places. In all the
investigations of this chapter these places are to be the same. They may be
called the essential singularities of the systems of factorial functions. We
require the surface to be dissected so that the places c1} ..., ck are excluded
and the surface becomes simply connected. This may be effected in a manner
analogous to that adopted in § 180, the places c1} ..., Ck occurring instead of
253]
DISSECTION OF THE SURFACE.
395
zlt ...,zk. But it is more convenient, in view of one development of the
theory, to suppose the loops of § 180 to be deformed until the cuts* between
the pairs of period loops become of infinitesimal length. Then the dissection
will be such as that represented in figure 9 ; and this dissection is sufficiently
Fig. 9.
well represented by figure 10. We call the sides of the loops (ar), (br), upon
which the letters ar, br are placed, the left-hand sides of these loops, and by
the left-hand sides of the cuts (7^, ..., (7^), to the places clt ..., c&, we mean
the sides which are on the left when we pass from A to d, ..., c^ respec
tively. The consideration of the effect of an alteration in these conventions
is postponed till the theory of the transformation of the theta functions
has been considered.
* These cuts are those generally denoted by clt ...,cp_,. Cf. Forsyth, Theory of Functions,
§181.
396 FUNDAMENTAL EXPRESSION OF FACTORIAL FUNCTIONS. [254
254. In connection with the surface thus dissected we take now a series
of 2p + k quantities
Xj, ...,\k, hl} ..., hp, gi,...,gp,
which we call the fundamental constants; we suppose no one of \1} ..., \k to
be a positive or negative integer, or zero ; but we suppose X, + . . . + \t to be
an integer, or zero ; and we consider functions
(1) which are uniform on the surface thus dissected, and have, thereon,
no infinities except poles,
(2) whose value on the left-hand side of the period loop (a,f) is
g-STriAj times the value on the right-hand side ; whose value on the left-hand
side of the period loop (h) is e27"^ times the value on the right-hand side,
(3) which*, in the neighbourhood of the place c;, are expressible in the
form t~^fa, where t is the infinitesimal at c; and fa is uniform, finite, and not
zero in the neighbourhood of the place c;,
(4) which, therefore, have a value on the left-hand side of the cut 7;
which is e~-ni*i times the value on the right-hand side.
Let «1( ..., OM, &, ..., f3N be any places; consider the expression
x,a x, a x, a x,a x,a x,akx,a
f= ^e11/?,, m+--- + lip,,, m - nai) m - ... - naj/> m - 2« [(A + ffJ vi +... + (hp + Hp) rp ] - S XjIIc., ,„
«/ t=l
wherein A is independent of the place x,
N-M=2\, (i),
*=i
SX being an integer (or zero), m is an arbitrary place, and Hl , ..., Hp are
integers. It is clear that this expression represents a function which is
uniform on the dissected surface, which has poles at the places al} ..., aM> and
zeros at the places /3j, ..., /3N, and that in the neighbourhood of the place c;
this function has the character required. For the period loop (a^) the
function has the factor e~Zvt^hi+a^ = e~Zwihi, as desired; for the period loop
(bi) the function has the factor eZwtK, where
r=l r=l
and this factor is equal to e2niyi if only
k
r=l
r=p
r=l
Gi being an integer.
* It is intended, as already stated, that the places cl, ..., c* should be in the finite part of the
surface and should not be branch places.
255] THE TWO SYSTEMS OF FACTORS. 397
It follows therefore that, subject to the conditions (i) and (ii), such a
function as has been described certainly exists.
Conversely it can be immediately proved that any such function must be
capable of being expressed in the form here given, and that the conditions
(i), (ii) are necessary.
Unless the contrary be expressly stated, we suppose the quantities
X^.-.jXjfc, hly...,hp, g1,...,gp always the same, and express this fact by
calling the functions under consideration factorial functions of the primary
system. The quantities e~2iri^, ..., e~-ni*k, e"2™'1*, ..., e~Zwihp, e27™^', ..., e27™^> are
called the factors. It will be convenient to consider with these functions
other functions of the same general character but with a different system of
fundamental constants,
Xt , . . . , Xj/, /*/, . . . , hp', g-t, ..., gpi
connected with the original constants by the equations
X; + X/ +1=0, hi + hi = 0, g{ + g- = 0 ;
these functions will be said to be functions of the associated system. The fac
tors associated therewith are the inverses of the factors of the primary system.
255. As has been remarked, the rational functions on the Riemann
surface are a particular case of the factorial functions, arising when the
factors are unity and no such places as cl5 ..., ck are introduced. From this
point of view the condition (i), which can be obtained as the condition that
Id log/, taken round the complete boundary of the dissected surface, is zero,
is a generalisation of the fact that the number of zeros and poles of a rational
function is the same, and the condition (ii) expresses a theorem generalising
Abel's theorem for integrals of the first kind.
Now Riemann's theory of rational functions is subsequent to the theory
of the integrals ; these arise as functions which are uniform on the dissected
Riemann surface, but differ on the sides of a period loop by additive
constants. In what follows we consider the theory in the same order, and
enquire first of all as to the existence of functions whose differential coefficients
are factorial functions. For the sake of clearness such functions will be
called factorial integrals; and it will appear that just as rational functions
are expressible by Riemann integrals of the second kind, so factorial functions
are expressible by certain factorial integrals, provided the fundamental con
stants of these latter are suitably chosen. We define then a factorial integral
of the primary system, H, as a function such that dH/dx is a factorial
function with the fundamental constants
398 FACTORIAL INTEGRALS. [255
thus dH/dx has the same factors as the factorial functions of the primary
system, but near the place d, dH/dx is of the form t~(*i+l} fa, where fa is
uniform, finite and not zero in the neighbourhood of d- Similarly we define
a factorial integral of the associated system, H', to be such that dH'/dx
is a factorial function with the fundamental constants
V + 1, . . . , V + 1, A/, . . . , V, ffi, ..., ffp,
or
— A] , . . . , A.£ , III, ... , lip, (/i , . . . , ffp 5
thus, if/ be any factorial function of the primary system, fdH'/dx is a
rational function on the Riemann surface, for which the places c1} .... Ck
are not in any way special. And similarly, if/' be any factorial function
of the associated system, and H any factorial integral of the primary
system, /' dH/dx is a rational function.
The values of a factorial integral of the primary system, H, at the two
sides of any period loop are connected by an equation of the form
where p is one of the factors e~-irihr, e^'UJr, and fl is a quantity which is
constant along the particular period loop. Near d, H is of the form
where At is a constant, fa is uniform, finite, and, in general, not zero in the
neighbourhood of C;, and (7f is a constant, which is zero unless Xt- -f- 1 be a
positive integer (other than zero), and may be zero even when X; + 1 is a
positive integer. After a circuit round d, H will be changed into
H = Ai + e~z^ rA* fa + Z-jriCi + Gi log t ;
thus, when Gt = 0,
H= He-zvi^ + Ai(l- er****),
and when d is not zero, and, therefore, \ + 1 is a positive integer,
H=H
in either case we have
where 7 = e~2iriki, and F is constant along the cut (7;).
Thus, in addition to the fundamental factors of the system, there arise,
for every factorial integral, 2p + k new constants, *2p of them such as that
here denoted by ft and k of them such as that denoted by F. It will be
seen subsequently that these are not all independent.
256] FACTORIAL INTEGRALS WITHOUT INFINITIES. 399
As has been stated we exclude from consideration the case in which any
one of \i, ..., At is an integer, or zero. Thus the constants Ci will not enter;
neither will the corresponding constants for the associated system.
256. Consider now the problem of finding factorial integrals of the
primary system which shall be everywhere finite. Here, as elsewhere, when
we speak of the infinities or zeros of a function, we mean those which are not
at the places d, ..., ck, or which fall at these places in addition to the poles
or zeros which are prescribed to fall there.
If V be such a factorial integral, dVjdx is only infinite when dx is zero
of the second order, namely 2p — 2 + 2n times, at the branch places of the
surface. And d V/dx is zero at x = oo , 2n times*. Thus, if N denote the num
ber of zeros of dVjdx which are not due to the denominator dx, or, as we may
say (cf. § 21) the number of zeros of dV, we have by the condition (i) § 254,
N + 2n = 2p - 2 + 2/i + 2 (X* + 1),
1=1
so that the number of zeros of dV is 2p - 2 + 2 (X; + 1).
Now let f0 denote a factorial function with the primary system of
factors, but with behaviour at a like J-to+D fa, where fa is uniform, finite,
and not zero at a. Then, if an everywhere finite factorial integral V
exists at all, Z, =f0-ldV/dx, will be a rational function on the Riemann
surface, infinite at the (say N0) zeros of /0, and 2w + 2p-2 times at the
branch places of the surface, and zero at the (say M0) poles of /0, and In
times at x = oo (beside being zero at the zeros of dV). Conversely a rational
f
function Z satisfying these conditions will be such that \ZfQdx is a function V.
Thus the number of existent functions V ivhich are linearly independent is at
least
provided this be positive. We are therefore sure, when this is the case, that
functions V do exist. To find the exact number, let F0 be one such ; then
if F be any other, dV/dV0 is a rational function with poles in the
2p - 2 +S(X+ 1) zeros of dV0; and conversely if R be a rational function
e
whose poles are the zeros of dVQ, the integral IRdV0 is a function F. Thusf
the number of functions V, when any exist, is (§ 37, Chap. III.)
*r, = p-l + 2(X + l) + <r+l,
* These numbers may be modified by the existence of a branch place at infinity. But their
difference remains the same.
t For the ordinary case of rational functions er + 1, as here defined, is equal to unity, and,
therefore, omitting the term S (\ + l), we have -a -p.
400 EXISTENCE OF FACTORIAL FUNCTIONS. [256
where <r + 1 is the number of linearly independent differentials dv, of ordinary
integrals of the first kind, which vanish in the 2p — 2 + 2 (X+l) zeros of the
differential dV0 of any such function V0. Since dV/dV0 is a rational
function, the number of differentials dv vanishing in the zeros of dV0 is the
same as the number vanishing in the zeros of dV. Since dv has 2p — 2 zeros,
a + 1 vanishes when 2 (A + 1) > 0.
Ex. For the hyperelliptic surface
the factorial integrals, V, having the same factors at the period loops as the root function
•J(x — a} (x — b), and no other factors, are given by
_ dx
*l(x-a)(x-V) (x, l)p_2 —
and -u3=p — 1. Here £=0 ; there are no places clt ... , ck.
257. The number <r + 1 is of great importance ; when it is greater
than zero, which requires 2 (X + 1) to be negative or zero, there are cr + 1
factorial functions of the associated system which are nowhere infinite.
For if V be an everywhere finite factorial integral of the primary system,
and dv1} ..., dvv+1 represent the linearly independent differentials of integrals
of the first kind which vanish in the zeros of dV, the functions
dV" '"'
whose behaviour at a place Ci is like that of TT^+T, &> where fa is uniform,
finite and not zero in the neighbourhood of a, namely of t^'fy, are clearly
factorial functions of the associated system, without poles. Conversely if K'
denote an everywhere-finite factorial function of the associated system, the
integral \K'dV is the integral of a rational function, and does not anywhere
become infinite. Denoting it by v, dv vanishes at the 2p — 2 + 2(A. + l)
k
zeros of dV as well as at the 0+2 A,/, = — 2 (A. + 1), zeros of K ' (cf. the
t=i
condition (i), § 254). Thus, to every factorial integral V we obtain <r + 1
functions K' ; and since, when <r + 1 > 0, the quotient of two differentials
dV, dV0 can* be expressed by the quotient of two differentials dv, dv0, we
cannot thus obtain more than <r + 1 functions K' ; while, conversely, to every
function K' we obtain a differential dv which vanishes in the zeros of any
assigned function V; and, as before, we cannot obtain any others by taking,
instead of V, another factorial integral V0.
* Cf. Chap. VI. § 98.
258] WHICH DO NOT BECOME INFINITE. 401
258. The existence of these everywhere finite factorial functions, K', of
the associated system can also be investigated a priori from the fundamental
equations (i) and (ii) (§ 254). These give, in this case,
- TI, p (hp + Hp), (iii)
and N=- 2 (Xr+l),
r=l
where Glt ..., Gp, Hl, ..., Hp are integers.
Hence no functions K' exist unless 2 (A, + 1) be a negative integer or be
zero ; we consider these possibilities separately.
When 2(X+1) = 0, it is necessary, for the existence of such functions,
that the fundamental constants satisfy the conditions
conversely, when these conditions are fulfilled, taking suitable integers
Hl} ..., Hp, it is clear that the function
wherein A is an arbitrary constant, and a, m are arbitrary places, is an
everywhere finite factorial function of the associated system, and it can be
immediately seen that every such function is a constant multiple of E0. If
then we denote the number of functions K' by 2 + 1 (to be immediately
identified with a + 1 ), we have, in this case, 2 + 1 = 1; and there are p
functions V, given by V=\E^dv, where dv is in turn the differential of
every one of the linearly independent integrals of the first kind ; it is easy to
see that every function V can be thus expressed. Thus, in the zeros of a
differential dV there vanishes one differential dv, so that a +1 = 1. Hence
o- + 1 = 2 + 1, and the formula w =p — l + 2(\ + l) + <r + l is verified.
When 2(\ + l) is negative and numerically greater than zero, and the
equations (iii) have any solutions, let t denote the number of linearlv in
dependent differentials dv which vanish in the places of one and therefore of
every set, &, ..., /9V, which satisfies these equations; then* the number of
sets which satisfy these equations is oo S-P+*} where s = — 2 (X + 1) ; thus the
quotient of two functions K' is a rational function with 2 + 1, =s—p+t + l
arbitrary constants, one of these being additive. This is then the number of
linearly independent functions K'. If K' be one of these functions, and
' Cf. § 158, Chap. VIII. ; § 95, Chap. VI.
B- 26
402 EXPRESSION OF FINITE FACTORIAL FUNCTIONS [258
dVi, ..., dvt denote the differentials vanishing in the zeros of K', it is clear
that the functions
fdvl rdvt
JK" "" IK'
are finite factorial integrals of the primary system, that is, are functions V ;
conversely if Fbe any finite factorial integral of the primary system, \K'dV
is an integral, v, of the first kind such that dv vanishes in the zeros of K'.
Hence the number t, which expresses the number of differentials dv which
vanish in the zeros of K', is equal to the number, CT, of functions V. But we
have proved that •& =p — I + '2t(\+l)+<T+l, and, above, that t=p -I— s+2+1.
Hence o- + l =2+ 1. .vl
Thus we have the results*: The number, cr + 1, of everywhere finite
factorial functions, K' , of the associated system is equal to the number of
differentials dv which vanish in the 2p — 2 + 2 (X + 1) zeros of any differential
dV; hence (§ 21, Chap. II.) <7 + 1 is less than p, unless 2 (X + 1) = - (2^> - 2).
Also, when a + 1 > 0, the number, vr, of everywhere finite factorial integrals,
V, of the primary system, is equal to the number of differentials dv which
vanish in the s, = - 2 (X + 1), zeros of any function K'. The argument by
which this last result is obtained does not hold whenf cr + 1 = 0. When
o- + 1 > 0, it follows that OT is not greater than p.
Similarly when s', = - 2 (X' + 1), = Sx, = - s - k, is > 0, we can prove, by
considering the primary system, that there are a' + 1 everywhere finite
factorial functions K of the primary system, where cr' + 1 is the number of
differentials dv vanishing in the 2p - 2 - 2X, =2p — 2 + s + k. zeros of any
differential dV "; and that, when cr'+l>0, the number &', of everywhere
finite factorial integrals, V , of the associated system is equal to the number
of differentials dv vanishing in the s' zeros of any function K. Hence
a' + 1 = 0 when s > 0, and, then, no functions K exist. When s = 0 we have
seen that there may or may not be functions K' ; but there cannot be func
tions K unless k = 0, since otherwise 2p — 2+s + k>2p— 2. And then the
existence of functions K depends on the condition whether the fundamental
constants be such that
is a function of the primary system or not, Hl} ...,HP being suitable integers,
namely whether there exist relations of the form
9i. + Gi+fa + H,} n, l + + (hp + Hp) ri>p = 0, (i = 1, 2, ..., p),
* Which hold for the ordinary case of rational functions, <r + l being then unity.
t In the case of the factorial functions which are square roots of rational functions of which
all the poles and zeros are of the second order, so that the places cl , . . . , ck are not present, and
the numbers g, h are half integers, we have cr=p - 1, a + 1 = 0.
260] BY FINITE FACTORIAL INTEGRALS. 403
where Glt ..., Gp are integers. In such case E0 is a finite factorial function
of the associated system.
On the whole then the theory breaks up into four cases (i) a + 1 = 0,
<r' + I = 0, (ii) a- + 1 > 0, a-' + 1 - 0, (Hi) o- + 1 = 0, a + 1 > 0, (iv) a + 1 = 1,
<T' + 1 = 1. Of these the cases (ii) and (iii) are reciprocal.
259. One remark remains to be made in this connection. When
v + 1 > 0 there are everywhere finite functions, K', of the associated system,
given (§ 257) by
dvi cfag dva+l
dV" dV" ' "' ~dV*
these have, at any one of the places clt ..., Ck, a behaviour represented by
that of t~*<f> ; hence the differential coefficients of these functions satisfy all
the conditions whereby the differential coefficients, dV'/dx, of the everywhere
finite factorial integrals of the associated system, are defined. Therefore* the
functions K' are expressible linearly in terms of the functions F/, ..., V'w>
by equations of the form
where the coefficients, \j, \ are constants.
Hence also the difference nr' — (a + 1) is not negative. This is also
obvious otherwise. For when <r + 1 > 0, — 2 (X + 1), =s, is zero or positive,
and cr + l>jp (§ 258), and, therefore, v - a; =p -(a- + 1) + er' + 1 + k + s,
can only be as small as zero when k = 0 = s, and a + 1 = p ; these are in
compatible.
Similarly, when a' + 1 > 0, the everywhere finite factorial functions of the
original system are linear functions of the factorial integrals Vl} ..., Vw.
It follows f therefore that of the «• periods of the functions F15 ..., FOT,
at any definite period loop, only -a - (a-' + 1) can be regarded as linearly
independent; in fact, a + 1 of the functions V1} ..., FOT may be replaced
by linear functions of the remaining w - (a-' + 1), and of the functions
Klt ..., KV'+I.
260. A factorial integral is such that its values at the two sides of a period loop of
the first kind are connected by an equation of the form u' = fiiU + Qiy its values at the two
sides of a period loop of the second kind are connected by an equation of the form
w'=/*'«M + Q't> and its values at the two sides of a loop (y{) are connected by an equation
of the form u'^yiU + Ti, where J 1^ = ^(1-^). Of the 2p+lk periods Qi} Q'f, I\ thus
* It is clearly assumed that K'i is not a constant ; thus the reasoning does not apply to the
ordinary case of rational functions.
t In the ordinary case of rational functions this number or - (<r + 1) must be replaced by p.
See the preceding note.
J § 255. The case where one of \j , ..., \k is zero or an integer is excluded.
26—2
404 DELATIONS AMONG THE PERIODS. [260
arising, two at least can be immediately excluded. For it is possible, by subtracting one
of the constants Alt ..., Ak from the factorial integral, to render one of the periods
TU ..., Tk zero; and by following the values of the factorial integral, which is single-
valued on the dissected surface, once completely round the sides of the loops, we find, in
virtue of y1y2 ••• yt=lj that
2 [Oi(l-/ii')-ai/(l-f*i)] = ri + y1ra + y1y2r3 + ... +y1y2...yk-lTk.
i=l
Thus there are certainly not more than 2p — 2+k linearly independent periods of a
factorial integral.
Suppose now that V is any everywhere finite factorial integral of the original system,
and Vi is any one of the corresponding integrals of the associated system. The integral
I Vd F/, taken once completely round the boundary of the surface which is constituted by
the sides of the period loops, is equal to zero. By expressing this fact we obtain an
equation which is linear in the periods of V and linear in the periods of F/. By taking i
in turn equal to 1, 2, ... , or', we thus obtain 07' linear equations for the periods of V,
wherein the coefficients are the periods of F1', . . . , V w>. As remarked above these coeffi
cients are themselves connected by <r + 1 linear equations ; so that we thus obtain at most
or' — (<r + l) linearly independent linear equations for the periods of F. If these are inde
pendent of one another and independent of the two reductions mentioned above, it follows
that the 2p + k periods of V are linearly expressible by only
2p
periods, at most. Now we have
and therefore
so that
Thus or-(<r'-f 1) is the number of periods of a function V which appear to be linearly
independent; and, taking account of the existence of the functions K^ ..., Ka'+\, this is
the same as the number of independent linear combinations of the functions Fn ... , FOT,
which are periodic*. But the conclusions of this article require more careful considera
tion in particular cases ; it is not shewn that the linear equations obtained are always
independent, nor that they are the only equations obtainable.
Ex. i. Obtain the lineo-linear relation connecting the periods of the everywhere finite
factorial integrals F, V, of the primary and associated system, which is obtained by
expressing that the contour integral I Vd V vanishes.
Ex. ii. In the case of the ordinary Kiemann integrals of the first kind, the relation
is identically satisfied, and further &=0. Thus the reasoning of the text does not holdt.
* We can therefore form linear combinations of the periodic functions V, for which the inde
pendent periods shall be 1, 0, . . . , 0 ; 0, 1, . . . , 0 ; etc., as in the ordinary case.
t In that case the numbers -a' — (ff + 1), 2p-2 + k, are to be replaced respectively by p and 2p.
See the note t of § 259.
262] RIEMANN-ROCH THEOREM FOR FACTORIAL FUNCTIONS. 405
261. We enquire now how many arbitrary constants enter into the
expression of a factorial function of the primary system which has M
poles of assigned position.
Supposing one such function to exist, denote it by FQ; then the ratio F/F0,
of any other such function to F, F0, is a rational function with poles at the
zeros of F0 ; conversely if R be any rational function with poles at the zeros
of F0, F0R is a factorial function of the primary system with poles at the
assigned poles of F0. The function R contains
N-p+ 1+h+l
arbitrary constants, one of them additive, where N is the number of zeros of
k
F0> so that N=M + 2 \r, and h + l is the number of differentials dv vanish-
r=l
ing in the zeros of FQ.
But in fact the number of differentials dv vanishing in the zeros of F0 is
the same as the number of differentials dV vanishing in the poles of F0, V
being any everywhere finite factorial integral of the associated system.
For if dv vanish in the zeros of F0> the integral ldv/FQ is clearly a factorial
integral, V, of the associated system without infinities, and such that dV
vanishes in the poles of F0 ; conversely if V be any factorial integral of the
associated system such that dV vanishes in the poles of FQ, the integral
lF0dV is an integral of the first kind, v, such that dv vanishes in the zeros
QfJt
Thus, the number of arbitrary constants in a factorial function of the
primary system, with M given arbitrary poles, is
k
M+ 2 \r-p+I+h + l, =N-p + i+h+ 1, =M-*r' + h + I+<r'+l
r=\
where N is the number of zeros of the function, and h + l the number of
differentials dV vanishing in the M poles*.
In particular, putting M=0, h . + 1 = *r' (cf. § 258), we have the formula,
already obtained,
k
o-'+l= 2 \r-p + !+-&'.
r=1
We can of course also obtain these results by considering the fundamental
equations (i) and (ii), § 254.
262. Hence we can determine the smallest value of M for which a
factorial function of the primary system with M given poles always exists.
* Counting the additive constant in the expression of a rational function, the last formula
holds in the ordinary case.
406 FACTORIAL FUNCTION WITH FEWEST ARBITRARY POLES. [262
When M = •&' + 1 it is not possible to determine a function V, of the
form
wherein A1} ..., A^> are constants, to vanish in M arbitrary places; and
therefore h + 1 = 0. Thus a factorial function of the primary system with
-OT' + 1 arbitrary poles will contain, in accordance with the formula of the
last Article,
fc
*r' + l + S Xr-p + 1, = <r'+2,
r=l
arbitrary constants.
When a' + 1=0, this number is 1, and the factorial function is entirely
determined save for an arbitrary constant multiplier. Hence we infer that
when a-' + 1 = 0 the smallest value of M is CT' + 1.
We consider in the next Article how to form the factorial function in ques
tion from other functions of the system. Of the existence of such a function
we can be sure a priori by the formulae (i) (ii) of § 254. Such a function
will have N = •&' + 1 + Sx, =p, zeros. They can be determined to satisfy the
equations (ii). Then an expression of the function is given by the general
formula of § 254.
When a' + I > 0, there are a + 1 everywhere finite factorial functions
Kl} ..., KS+I, of the primary system, and the general factorial function with
TV' + 1 poles is of the form
where \ , ..., X^+j are constants, and F is any factorial function with the
assigned poles. In this case also there exist no factorial functions with
arbitrary poles less than is' + 1 in number ; the attempt to obtain such
functions leads* always to a linear aggregate of Klt ...,^T<r'+1.
263. Suppose that a' + 1 = 0 ; we consider the construction of the
factorial function of the primary system with •&' + 1 arbitrary poles.
Firstly let a + 1 > 0, so that there are cr + 1 everywhere finite functions,
K', of the associated system, and cr + 1 differentials dv vanish in the
k k
2p — 2 + 2 (Xf + 1) zeros of any differential dV. Hence s, = — 2 (Xr+l),
r=l r=l
is greater than zero or equal to zero. We take first the case when s > 0.
*
Then ^'=p — I— 2 \ =p — 1 + s + k, and it is possible to determine a
r=l
rational function with poles at -57'+ 1 =p + s + k arbitrary places. This
function contains s + k + 1 arbitrary constants, one of these being additive.
It can therefore be chosen to vanish at the places clt ..., ck> and will then
* For J/ = •or' - r, we shall have h + 1 = r, and, therefore, M - &' + h + 1 + a' + 1 = <r' + 1.
264] METHODS FOR CONSTRUCTING THIS FUNCTION. 407
contain at least, and in general, s+I arbitrary constants. Taking now any
everywhere finite factorial function K' of the associated system, let the
rational function be further chosen to vanish in the s zeros of K' ; then the
rational function is, in general, entirely determined save for an arbitrary
constant multiplier. Denote the rational function thus obtained by R.
Then RjK' is a factorial function of the primary system with the -a' 4- 1
assigned poles, and is the function we desired to construct. And since the
ratio of two functions K' is a rational function, it is immaterial what function
K' is utilised to construct the function required.
This reasoning applies also to the case in which <r + l >0, s = 0, unless
also k = 0. Consider then the case in which a + 1 > 0, s = 0 and k = 0.
There is (§ 258) only one function K', of the form
-a+ ...... + (hp + Hp) vx; "•]
,
or or -f 1 = 1 ; and E0~l is a function of the primary system without poles.
Thus a' + 1 = 1, and the case does not fall under that now being considered,
for which a + I = 0. The value of w' is p, and the factorial function with
•BT' + 1 arbitrary poles is of the form (F + C) E0, where F+ C is the general
rational function with the given poles.
Nextly, let cr + 1 = 0, as well as <r' + 1 = 0. Then there exist no functions
K' and the previous argument is inapplicable. But, provided OT' + 1 <fc 2, we
can apply another method, which could equally have been applied when
o- + 1 > 0. For if P be the factorial function of the primary system with
is' + 1 assigned poles, and V be one of the CT' factorial integrals of the
associated system, and v be any integral of the first kind, P —, is a rational
function whose poles are at the w' + 1 poles of P and at the 2p — 2 zeros of
dv. Conversely, if R be any rational function with poles at these places
(c£§37, Ex. ii. Chap. III.), and zeros at the 2p - 2 - 2\ zeros of dV, R\dJ
I civ
is the factorial function required. It contains at least
arbitrary constant multiplier.
In case *r' + 1 < 2, so that «•' = 0, S\ = p - 1, there are no functions V,
and we may fall back upon the fundamental equations of § 254. In this case
the least number of poles is 1.
264. Consider now the possibility of forming a factorial integral of the
primary system whose only infinities are poles. We shew that it is possible
to form such an integral with <r + 2 arbitrary poles, and with no smaller
number.
408 CONSTRUCTION OF THE FACTORIAL INTEGRAL [264
Suppose G to be such a factorial integral, with <r + 2 poles, and, under the
hypothesis CT > 0, let V be an everywhere finite factorial integral, also of the
primary system. Then dG/dV is a rational function, with poles at the
2p - 2 + 2(A + 1) zeros of dV, and poles at the poles of G] near a pole
of G, say c, the form of dG/dV is given by
where t is the infinitesimal for the neighbourhood of the place c, the
quantities 0, A, B are constants, and DCV denotes a differentiation in regard
to the infinitesimal ; this is the same as
-r — E\—- + - ° + terms which are finite when t
= 0 ,
where E = — C/DCV. Thus dG/dV is infinite at a pole of G like a constant
multiple of
. nr«*>a DC*V x,a
y = DCl-C - p~y I C ,
a being an arbitrary place.
Conversely if R denote a rational function which is infinite to the first
order at the zeros of d V, and infinite in the a- + 2 assigned poles of G like
functions of the form of ty, \RdV will be such a factorial integral as desired.
J
Now R is of the form (§ 20, Chap. II.)
e> a , ft FT-) -p*. a DXl V x, a\
L "*i ' ' J
|~ ,a D*<r+2V x,a 1
wherein a is an arbitrary place, e1} ..., er denote the zeros of dV, x-^, ..., xy+n
denote the assigned poles of G, and A, A1} ..., Ar, Blt ..., Bv+.2 are constants;
the period of R, in this form, at a general period loop of the second kind, is
given by
-O-iiij \@i) "T" T -^-r^^i \^r) ' -^
where ^(x), ..., Qp(x) are as in § 18, Chap. II., and this must vanish for
i = l, 2, ..., p. Now (§ 258) in the places el} ..., er there vanish a + 1 linear
functions of f^ (x), ..., O^ (a;). Thus, from the conditions expressing that the
periods of R are zero, we infer a + 1 linear equations involving only the
constants Blt ..., Ba+2, which, since the places oelt ..., ov+a are arbitrary, may
be assumed to be independent. From these cr + 1 equations we can obtain
265] WITH FEWEST ARBITRARY POLES. 409
the ratios Bl:Bz: ...... : B^2. There remain then, of the p equations
expressing that the periods of R are zero, p — (cr + 1) independent equations
containing effectively r + 1 unknown constants. Thus the number of the
constants Alt ..., Ari Blt ..., B^ left arbitrary is r+ 1 -p + a- + I, which is
equal to 2p -2 + 2 (X + !) + !— p + a + 1 or tzr, and the total number of
arbitrary constants in R is CT + 1. Thus we infer that, on the whole, G is of
the form*
where [G] is a special function with the cr + 2 assigned poles, multiplied by
an arbitrary constant, and Glt ..., G^ , G are arbitrary constants. And this
result shews that cr -f 2 is the least number of poles that can be assigned for
G. The argument applies to the case when cr + 1 = 0 provided that w > 0.
The proof just given supposes w > 0 ; but this is not indispensable.
Let f0 be a factorial function with the primary system of multipliers but
with a behaviour at the places c; like t~(^+1)(j>ii where fa is uniform, finite
and not zero in the neighbourhood of a. Then if, instead of IRdV, we
consider an integral IRfodv, wherein dv is the differential of any Riemann
integral of the first kind, and R is a rational function which vanishes in the
(say M) poles of /0, and may become infinite in the zeros of dv and the
(say N) zeros of f0i we shall obtain the same results. It is necessary to
take N> 1 (cf. § 37, Ex. ii. Chap. III.).
265. Another method, holding whether -sr = 0 or not, provided a + 1 > 0,
may be indicated. Let K'(x) be one of the everywhere finite factorial func
tions of the associated system. Consider the function of x,
a, c, 7 being any places and A a constant ; when x is in the neighbourhood
of the place c it is of the form
l_ \ X _l t_ — I
where t is the infinitesimal in the neighbourhood of the place c, and terms
which will lead only to positive powers of t under the integral sign are
omitted ; this is the same as
* In the ordinary case of rational functions, where V is replaced by a Rieinann normal inte
gral v, the coefficients of #,, ... , B<r+2, in the expression for the general period of R, vanish for
one value of i, namely when V=v(. Thus o- + l( = l) pole is sufficient to enable us to construct
the factorial integral ; it is the ordinary integral of the second kind.
410 SIMPLIFICATION OF THE INTEGRAL [265
hence if A be DK\c)/K'(c), the function i/r is infinite at c like - - ™
- ™ —
t K (c)
A
At the place 7 the function ty is infinite like - JJTT-^ log ty, where ty is
the infinitesimal in the neighbourhood of the place 7.
Putting now M*]* = 1^' a + ^ IlJ ", consider the function
K (C)
A */ix> a , n x>a -n x
+ A*+*M*r+z,v + B^ +••• + BPVP
where a, 7 are arbitrary places and A^..., Aa+t, Bl}..., Bp are constants,
subject to the conditions
(i) that
AlDxMxx["y + ...... + A9+iD,M2^ty + BA(x)+ ...... + Bpflp(a>)
vanishes at each of the - 2 (X + 1) zeros of K'(x),
(ii) that
DK'(*d DK'(x<r+.)
1 ZW' A'+9 K'(a,.+t) = °'
the first condition ensures that G(x} is finite at the zeros of K'(x), the
second condition ensures that G(x) is finite at the place 7. If we suppose*
vi ,-•> v&a to be those integrals of the first kind whose differentials
vanish at the zeros of K'(x) (§ 258), the conditions (i) will involve only the
constants A1} ..., Aa+z, B^+l, ..., Bp> and if these conditions be independent
these a- + 2 + (p — w) coefficients will thereby be reduced to
>sr + 2(\ + l), = 2 ;
thus, if the condition (ii) be independent of the conditions (i), the number
of constants finally remaining isw + 2 — l=«r + l, and the form of G(x} is
[G] + C1V1 + ...... + CWV^ + C
as before.
Ex. Prove that, when s, = - 2 (X + 1 ), is positive, we have
266. The factorial integral of the primary system with o- + 2 arbitrary
poles can be simplified. If a;1, ..., av+2 be the poles, its most general form
may be represented by
* This is to simplify the explanation. In general it is tzr linear combinations of the normal
integrals, whose differentials vanish in the zeros of K'(x). The reduction corresponding to that of
the text is then obtained by taking or linear combinations of the conditions (i).
267] IN ANALOGY WITH THE ORDINARY CASE. 411
where E, Elt ..., E^, C are arbitrary constants. Near a place clt one of the
singular places of the factorial system, the integral will have a form
represented by A^ + £~A> <f> ; we may simplify the integral by subtracting
from it the constant A^, the consequence is that the additive period
belonging to the loop (7^ is zero ; further there is one other linear relation
connecting the additive periods of the integral, which is obtainable by
following the value of the integral once round the boundary of the dissected
surface (cf. § 260). Thus the number of periods of the integral is at
most 2p — 2 + k. We suppose the additive periods of the functions
G (xl , . . . , av+2), y\ , •••, Vw , at the loop (71), to be similarly reduced to zero ;
then the constant C is zero. The linear aggregate E1V1 + ...... + E^ V^
may be replaced by an aggregate of the non-periodic functions Klt ..., K^+i,
and Br-(<r' + l) of the integrals F,,..., FOT, so that the integral under
consideration takes the form
EG fa, ..., av+a) + CM + ... 4- CW-((r<+1) FOT_((r.+1)
where C1} ..., C^-^'+D, Flt..., F^+i are constants. We can therefore, pre
sumably, determine the constants Cl}..., C^_ („.•+!), so that sr — (a-' + 1) of
the additive periods of the integral vanish. The integral will then have
2p — 2 + k — (w — a-' — 1), = CT' — (a- + 1), periods remaining, together with one
period which is a linear function of them. A particular case* is that of
Riemann's normal integral of the second kind, for which there are p periods.
As in that case we suppose here that the period loops for which the additive
periods of the factorial integral shall be reduced to zero are agreed upon before
hand. We thus obtain a function
wherein F, Fly ... , F^+i are arbitrary constants, and G^ (a-j. ..., xff+.i) has
additive periods only at CT' — (cr + 1) prescribed period loops, beside a period
which is a linear function of these. We may therefore further assign <r' + 1
zeros of the integral and choose F so that the integral is infinite at a\
like the negative inverse of the infinitesimal. When the integral is so
determined we shall denote it by F(a;1, a;.,, ... , av+g). The assigned zeros are
to be taken once for all, say at aly ..., <v+1.
267. The factorial function of the primary system with -or' + 1 assigned
arbitrary poles can be expressed in terms of the factorial integral of the
primary system with <r + 2 assigned poles. Let xly ..., xw>+l be the assigned
poles of the factorial function. Then we may choose the constants C\ , ...,
G-aj'-a, so that the •&' — (a- + 1) linearly independent periods of the aggregate
are. all zeros. The result is a factorial function with xl , . . . , av+1 as poles,
* Of the result. The reasoning must be amended by the substitution of p, 2p for -as' - (a + 1)
and 2p - 2 + k respectively. Cf. the note t of § 260.
412 EXPRESSION OF THE FUNDAMENTAL FACTORIAL FUNCTION [267
which vanishes in the places a1} ..., cv+1. Or, taking arbitrary places
d1 , ..., da+l we may choose the constants El} . . . , EW>+1 so that the w' — (a + 1)
linearly independent periods of the aggregate
ElY(xl, dl} ..., d<T+1) + E2T(x2,d1, ...,d.+l) + .
are all zero, and at the same time the aggregate does not become in
finite at di, ...,dv+1. Then the addition, to the result, of an aggregate
F1K1 + ...... + Fa'+1K0'+1, wherein Fl} ..., F^+l are arbitrary constants, leads
to the most general form of the factorial function with xl} ..., xw'+l as poles.
For the sake of defi niteness we denote by ty (x ; z, tlt . . . , tw>} the factorial
function with poles of the first order at z, t1} ..., t&>, which is chosen so that
it becomes infinite at z like the negative inverse of the infinitesimal, and
vanishes at the places a^, ..., aa>+1. A more precise notation would be*
^r (x, alt . . . , <v+1 ; z, t1} . . . , t&>). This function contains no arbitrary constants.
Denoting this function now, temporarily, by -fy, and any everywhere
finite factorial integral of the inverse system by V, the value of the integral
fydV, taken round the boundary of the dissected surface formed by the
sides of the period loops, is equal to the sum of its values round the poles
of i|r. Since ^dV/dx is a rational function the value of the integral taken
round the boundary is zero. Near a pole of ^r, at which t is the infinitesimal,
the integral will have the form
where D denotes a differentiation. Thus the value obtained by taking the
integral round this pole is A (DV). If then the values of A at the poles
Fn ..., FOT' be denoted by Alt . .., A^, we have, remembering that the
value of A at z is — 1, the -nr' equations
A,
where F/, ..., V'w' are the •or' everywhere finite factorial integrals of the
associated system, (DF/)r denotes the differential coefficient of F/ at tr, and
(DVt\ denotes the differential coefficient at z. Thus, if wr(x) denote, here,
the linear aggregate of the form
wherein the constants Elt ..., E^> are chosen so that wr(tr) = 1 and wr(ts) = 0
when ts is any one of the places ^, ..., t^> other than tr, we have Ar = o)r(z).
Hence we infer by the previous article (§ 266) that ty(x\ z,t1} ..., tw<) is
equal to
T(z,dlt ...,C?(T+])-WI(^) F(^, d1} ..., dc+1)- ...... -a)w>(z)r(t^'ydl) ...,d0+1),
* Cf. § 122, Chap. VII. etc.
268] BY MEANS OF THE FUNDAMENTAL FACTORIAL INTEGRAL. 413
where dl} ..., da+l are arbitrary places. For these two functions are infinite
at the places z,tlt ..., tw> in the same way and both vanish at the places
ttl, ..., &0'+\'
As in the case of the rational functions, the function ty (#; z, ti, ..., £OT')
may be regarded as fundamental, and developments analogous to those given
on pages 181, 189 of the present volume may be investigated. We limit
ourselves to the expression of any factorial function of the primary system by
means of it. The most general factorial function with poles of the first
order at the places ^ , . . . , zm may be expressed in the form
where A1} . .., Au, Blt ..., B^+l are constants. The condition that the
function represented by this expression may not be infinite at tr is
A1a>r(z1)+ ...... +Ajrur(zM)=0;
in case the ta equations of this form, for r = 1, 2, ..., CT', be linearly indepen
dent, the factorial function contains M + a-' + 1 — m' arbitrary constants;
but if there be h + 1 linearly independent aggregates of differentials, of the
form
C1dVl' + ...... + CW'dFV,
which vanish in the M assigned poles, then the equations of the form
A1a>r(z1)+
are equivalent to only vr' — (h + l) equations, and the number of arbitrary
constants in the expression of the factorial function is M + cr' + 1 — vrf + h + 1,
in accordance with § 261.
Ex. i. Prove that a factorial integral of the primary system can be constructed with
logarithmic infinities only in o- + 2 places, but with no smaller number.
Ex. ii. If the factorial integral Q (x^ x%, ... , #<r + 2) become infinite of the place xt like
j , where t is the infinitesimal at x^ prove, by considering the contour integral \GdKr',
where A',.' is one of the a- + 1 everywhere finite factorial functions of the associated system,
and 0 denotes G (xlt x2, ... , #<r + 2)j the <r + 1 equations
^
D denoting a differentiation. From these equations the ratio of the residues R1, R9, ...,
Ra + 2 can be expressed.
268. The theory of this chapter covers so many cases that any detailed
exhibition of examples of its application would occupy a great space. We
limit ourselves to examining the case p = 0, for which explicit expressions can
be given, and, very briefly, two other cases (§§ 268 — 270).
414 THE GENERAL THEORY TESTED [268
Consider the case p = 0, k = 3, there being three singular places such as
have so far in this chapter been denoted by clt C2, ..., but which we shall
here denote by a, /3, 7, the associated numbers* being \l = — 3/2, X2 = — 3/2,
\2 = — 2. At these places the factorial functions of the associated system
behave, respectively, like t~^(j)ly t~^2, t~l<f>z, and the difference between the
number of zeros and poles of such a function is N' — M' = — 2 (X + 1)= 2.
Thus there exist factorial functions of the associated system with no
poles and two zeros. By the general formula of § 254, replacing 11*'" by
(CC ~~ G I Ct ~~ G\
- I - 1 , the general form of such a function is found to be
as-yi a - 7/
+ Bx + C
=
(x-rix-ax-
and involves three arbitrary constants, so that a + 1 = 3. In what follows
K' ' (x) will be used to denote the special function l/(x — y)(a; — OL^(X — /3)i
The difference between the number of zeros and poles of factorial functions
of the primary system is N — M = — o ; hence M=Q is not possible, and
a-' + 1 = 0. Further
OT , = p _ 1 + £ (\ + 1) + <r + 1, =-1-2 + 3 = 0,
or', =p-l-2\ + <r' + l ,=-1 + 5 =4,
and the factorial function of the primary system with fewest poles has
•BT'+ 1 = 5 poles, as also follows from the formula N— M = — 5. This function
is clearly given by
(x-
(x — x-i) (x — x2) (x — x3) (x — #4) (x — xs) '
Putting
i/r (x} = (x-a.)(x- 0) (x - 7), f(x) = (x- tfj) (as - x2) (x - x3) (x - xt) (x - #5),
<£ O) = DK' (x)IK' (x} = -[(oc- 7)-1 + i (x - a)"1 + £ (x - P)'1],
and putting A,,- = ty («»)/./" O^f)* where i is in turn equal to 1, 2, 3, 4, 5 and
f'(x} denotes the differential coefficient of f(x), it is immediately clear that
P(x) is infinite at x^ like Xj/(# — x-^ K' (x^ It can be verified that
2\! = 0, i^^^l, i^1\1</>(a;1) = 0, ^scl-\l(j)(xl}= -2, SX10(^1) = 0,
111 i i
and these give
1 X, [1 + x^ (x,)} = 0, i\, [2^ + xty (x,)] = 0.
i i
The factorial integral G, of the primary system, with <r + 2 = 4 poles,
T, £, 77, £, is (§ 265) given by
* It was for convenience of exposition that, in the general theory, the case in which anjr of the
numbers X,, ... , \k are integers, was excluded.
268]
BY A CASE IN WHICH THE DEFICIENCY IS ZERO.
415
where the sign of summation refers to r, %, 77, £ and the constants Al} A2,
A3, AI are to be chosen so that (i) the expression
Arf (r) + A2$ (f ) + A3<j> (77) + A<4> (£)
is zero, this being necessary in order that G(r, f, 77, £) may not become
infinite at the place c, and (ii) the expression
vanishes to the fourth order when x is infinite ; the expression always
vanishes to the second order when x is infinite ; the additional conditions are
required because K' (x) is zero to the second order when x is infinite.
Taking account of condition (i), we find, by expanding in powers of - , that
JO
the condition (ii) is equivalent to the two
1 Al [1 4- T(f) (T)] = 0, I A, [2r + T2<£ (T)] = 0.
i i
Thus, introducing the values of A1} ..., A4 into the expression for
G (T, g, 77, £), we find, by proper choice of a multiplicative constant,
1 $ (T) /t\ t \ (
±i (£)> (^X (
(a; - r)2 x - r
1 + T </> (r),
....ax
in which the second, third and fourth columns differ from the first only in
the substitution, respectively, of £, 77, £ in place of T.
The factorial integral G(r, g, 77, £) thus determined can in fact be
expressed without an integral sigo. For we immediately verify that
is equal, save for an additive constant, to
T- + 1 + r0 (r) + i \x - 7 - i (a + £)} 0 (T)
(^a)- ft)
) - {7
(T)) + *7 (a
x log \x -
I
<#> (T)
T — a T —
xlog
a; - /8) (r -a) + V(a? - a) (T -
Va;— T
416
THE GENERAL THEORY TESTED
[268
and, by the definition of <f> (#), the coefficient of the logarithm in the last line
of this expression is zero ; if we substitute these values in the expression
found for G (r, %, rj, £) we obviously have
7 — T
(j) (T), . , . , .
-f constant,... (2),
where the second, third and fourth columns of the determinant differ from
the first only in the substitution, in place of r, respectively of £ V, £ We
proceed now to prove that this determinant is a certain constant multiple of
(x - a) (x - ft) (x - fi)/(x -T)(X- f ) (x -^(x- £), where n is determined by
4-
T
If we introduce constants, A, B, C, A', B', 6", depending only on a, ft, 7,
defined by the identities
2
Co? + Bx + A = n(x
4
we can immediately verify that
A(j> (a-) + B [1 + x$ (x)] + C [2a?
x — a
' [1+
and hence that
CC — T
thus
7 — T
' (a — r) (/3 — T) x — r '
7~T ! m f,
, w2f -x- -. .
(a — T) (/3 — T) x — T
<£(r), . , . ,
1-f T$ (T), . , . ,
-f constant,
...(3)
268] BY A CASE IN WHICH THE DEFICIENCY IS ZERO. 417
now it is clear from the equation (2) that G (T, g, i), £)/V(# — a) (x — /?) is of
the form (x, l\j(x — r)(x — £) (x — rj)(x — £), where (x, I), denotes an integral
cubic polynomial; and since l/K'(x) vanishes when a; = y, it follows from
the equation (1) that the differential coefficient of G(T,%,V), £) vanishes
when a" = y. Hence we have
where //. is such that the differential coefficient of this expression vanishes
when x = y, and has therefore the value already specified, L is a constant
whose value can be obtained from the equation (3) by calculation, and M
is a constant which we have not assigned. In the neighbourhood of the
place a, G (r, £, 77, £) has the form M + L(x- a)- [\ + p (x - a) + v (x-&f + ...},
and similarly in the neighbourhood of the place @. In the neighbourhood
of the place y, G (r, g, 77, £) has the form
N + (x - 7)- [V + // (a -7) + v'(x- 7)2 + ...... ].
where N is a constant, generally different from M.
In the general case of a factorial integral for p=0, k=3, the behaviour of the integral
at a, /3, y is that of three expressions of the form
provided no one ofX + l,/t + l,j/+l be a positive integer; herein one of the constants
A, B, C may be taken arbitrarily and the others are thereby determined. The factorial
integral becomes a factorial function only in the case when all of A, B, C are zero.
We have seen that the factorial function of the primary system with
fewest poles has 5 poles ; let them be at T, rl} %, 97, £; then, taking G (r, %, rj, £)
in the form just found, the factorial function can be expressed in the form
P (x} = CO (r, fc rj, ^ + CtG (TU fc 77, 0 + D,
when the constants C, Gl} D are suitably chosen.
For clearly D can be chosen so that the function P (x) divides identically
by (x — af-(x — ftf-. It is then only necessary to choose the ratio G : Cl}
if possible, so that the function P (x) divides identically by (x — y)-. This
requires only that
X-T l X - T, ^ (X - T) (X - Tj) '
where p is a constant, or that the expression
B. 27
418 THE GENERAL THEORY TESTED [268
divide by (x - 7)-. Thus C : C, = - (7 - T) (7 - /*,) : (7 - /*) (7 - TJ), and
27 - /A - TI _ 27 - /*! - r
(7 - fl) (7 - Tj) (7 - /ij) (7 - T) '
or
7 - /* 7-r 7 - /*i 7 — Ti
this condition is satisfied ; both these expressions are by definition equal to
J_ J_ _J__a' J • J
«• I t» '2
From the theoretical point of view it is however better to proceed as
follows— Let the poles of P (x} be at ar, , . . . , <r5. Then P (a-) can be expressed
in the form
Px = ClG(xl, £ 77, 0 + 0,0 (a?3> £ 77, 0+ ...... +CBG(ar0, fc 17, 0 + C,
the constants C, Glt C.,, ..., Cs being suitably chosen. This equation requires,
by equation (1),
+
0, #, J7, S
r [1 + xr<f> (#,)], 1 4- ^ (|), 1 + it (*)), 1 + ^ (D
i
5
wherein A (£, 77, £) is the minor, in the determinant occurring in equation (1),
of the first element of the first row, and E = (x - % )~2 + </> (£) (x - I)-1,
F= (x - r})-" + <f> (77) (x - rj)~l, G = (x- ^)~2 + <f> (^) (^ - f)-1. If now we take
C'j , .... (75 so that
55 5
^ /"' *Jv / /v» \ — rt ^? t r 1 l /y /"fk / *v* i I — C\ ^^ I , \ s nr I ^ *• /T\ i "T* i I ~~* 1 1
— * • CD \ w->* I — V/j .^^l^'i' I -L "T" vUf^r \ T/ I — ^? ^*\J '-p \£t*Asf i^ tAsj* \L/ y**/'j*/J ^^ ^j
11 i
this leads to
/ /yi /y» /y» /y> \ I. f] 7j /Jt / //» //» /V» 0" I
l w/j j tX/2 j ^s j *^4/ T^ ^^5-*-^ ^* V 1 ) 2 ? 3 3 5/J
77, g
and the solution can be completed as before.
There are TS' = 4 everywhere finite factorial integrals of the associated
dV .
system ; if V be one of these, then by definition, -^ is a factorial function
2(59] BY A CASE IN WHICH THE DEFICIENCY IS ZERO. 419
which has at a the form (x — a)~3<£, and similarly at ft, and has at 7 the
form (x — y)-2<f>. Further dV'jdx is zero to the second order at x = x .
Hence we have
(x, 1)3 cfa
_ _
Jx-
and dV has 2p — 2 — t\ = — 2 + 5 = 3 zeros.
Thus V can be written in the form
dx La? + MX +
= NK' (x) + Mr/ O) + £#,' (a?) + ^ F0',
where JV, Jlf, Z, .R are constants, K' (x), KJ (x), K2' (x) are particular, linearly
independent, everywhere finite factorial functions of the associated system,
and FO' is a particular everywhere finite factorial integral of the associated
system.
Ex. i. In case of a factorial system given by p = 0, k = 2, Xt= -f, X2= — f, prove that
<r-M=2, a-' +1=0, or = 0, or' = 2 ; prove that the factorial function of the primary system
with fewest poles is P (x} = (x-aft (x - @$l(x - x^) (x-x^ (x-xj) ; obtain the form of the
factorial integral of the second kind of the primary system with fewest poles, and prove
that it can be expressed in the form AP(x) + B ; and shew that the everywhere finite fac
torial integrals of the associated system are expressible in the form (Ax+B)/\/(x — a) (x — ft),
their initial form being
(Ax+E)dx
Ex. ii. When we take p = Q and k, =2n + 2, places clt ..., c2tt+2, and each X=-i,
prove that the original and the associated systems coincide, that a + 1 = o-' + 1 = 0, w = •&' = n,
that the everywhere finite factorial integrals, and the integral with one pole are respec
tively
fo!)-irfr /T/(a) +i^'(a)1 dx
J -JfV) JL(.'-«)2 + *^^Jx//^)'
where f(x) = (x-cl) ...... (^-c.2)l + 2). The factorial function with fewest poles is
^/ (*)/(-r, l)» + i; express this in the form
•JfW _»+> (T ffa) ../'(oi)! dx Lr(^l)»-x , ,
/_, i\ = 2 AI I - ----- a+f — . — + I — V- —- dx+ constant,
(x, l)n + 1 ,-=i j\_(x-ai}i - A'-aJv//(.r) J v(/(.^)
a,, ... , nn + 1 being the zeros of (x, 1),, + 1, and determine the 2?i + l coefficients on the right-
hand side.
269. One of the simplest applications of the theory of this chapter is to
the case of the root functions already considered in the last chapter ; such a
function can be expressed in the form e+, where
27—2
420 EXAMPLE OF RADICAL-FUNCTIONS. [269
where /3l5 ..., /3N are the zeros, al, ..., a.N the poles, h; is a rational numerical
fraction, Hi is an integer, and 7 is an arbitrary place. The singular places,
Cj, ...,Cfc are entirely absent. The zeros and poles satisfy the equations
expressed by
where Gl} ..., Gp are integers; and since, if m be the least common denomi
nator of the 2p numbers g, h, the rath power of the function is a rational
function, there is no function of the system which is everywhere finite,
and the same is true of the associated system. Hence er + l=0 = o-' + l,
•sr = OT' =p — 1 ; thus the function of the system with fewest poles has
p poles, and every function of the system can be expressed as a linear
aggregate of such functions (§ 267. Cf. § 245, Chap. XIII.).
Ex. i. Prove that when the numbers g, h are any half-integers, the everywhere finite
integrals of the system are expressible in the form
„ [
V=j
civ?-1.
- S
where v is an arbitrary integral of the first kind, (f> is the corresponding (^-polynomial,
and 3>t-, <&i are (^-polynomials with p — 1 zeros each of the second order (cf. § 245,
Chap. XIII.). It is in fact possible to represent any half-integer characteristic as the
sum of two odd half-integer characteristics in 2p~2(2p~1-l) ways.
Ex. ii. In the hyperelliptic case, when the numbers g, h are any half-integers, prove
that the function of the system with 07' + 1 =p poles is given by
\f >
)}
where the places (xlt y±), ... are the poles in question,
^ (#) = (#- #1) ... (% - xp), V/ Or) = dty (x}jdx, u = (x — a) (x — 6),
and a, b are two suitably chosen branch places*, and ui = (xi-a)(xi — 'b'). Shew that in
(T (it — - 1) l\ "Wi
the elliptic case this leads to the function — —. - r— e~v ("~").
cr (u - V)
270. In the case in which the factors at the period loops are any
constants, the places c1? ...,ck being still absent, it remains true that the
number of zeros of any function of the system is equal to the number of
poles; but here there may be an everywhere finite function of the system,
and there will be such a function provided
gi + Ti,i h,+ ...... +Tilphp = -[Gi + Tiil H, + ...... + Titp Hp], (i = l, 2, ...,p)
in which G1} ..., Hp are integers, the function being, in that case, expressed by
* For the association of the proper pair of branch places a, b with the given values of the
numbers g, h, compare Chap. XI. § 208, Chap. XIII. § 245, and the remark at the conclusion of
Ex. i.
270] A MOKE GENERAL CASE. 421
then E~l is an everywhere finite function of the associated system, and
a + I = a-' + 1 = 1, ur = CT' = p. It is not necessary to consider this case, for
it is clear that every function of the system is of the form ER, R being a
rational function.
When a- + 1 = a-' + 1 = 0 we have or = p — 1 = CT'. Then every function
of the system can be expressed linearly by means of functions of the system
having p poles. If #, , . .., xp be the poles of such a function and z1} ..., zv the
zeros, and the relations connecting these be given by
»*»*>+ + vz<»x»=f/+G + T(h + H).
There is beside the expression originally given, a very convenient way of
expressing such a function, whose correctness is immediately verifiable,
namely
wherein
and m, mly ...,mp are related as in § 179, Chap. X. Omitting a constant
factor this is the same as
r v /> j »
<s) (u)
since the difference between the values of the logarithm of <f> (u) at the two
sides of any period loop is independent of u, and of x, it follows that
o p,
^- log </> (u) is a rational function of x, and that ^- log <j> (u) is a periodic
function with 2p sets of simultaneous periods ; thus the function <£ (a)
satisfies linear equations of the form
^\.7 *o
W = RlJ> 55^ -•B«* <M = 1,2,...,,C),
where R, R,j are rational functions of x, and 2/;-ply periodic functions of u.
given* by
The 2/» constants a, X can be chosen so that
satisfies the equations <£ (M + 2«) = zl^(?«), </> (M + 2»') = ^l'0(?0. whore .4, J' each represents
p given constants, and the notation is as in § 189, Chap. X.
* Cf. Halphen, Fonct. Ellipt., Prem. Part. (Paris 1886), p. 235, and Forsyth, Theory of Func
tions, pp. 275, 285, for the case p = l. By further development of the results given in Chap. XI.
of this volume, and in the present chapter, it is clearly possible to formulate the corresponding
analytical results for greater values of j>.
422 A THETA FUNCTION OF GENERAL ARGUMENT [271
271. We have seen (§ 261) that the number of arbitrary constants
entering into the expression of a factorial function of the primary system
with given poles is N— jj+l+/i + l, =R say, where N is the number of
zeros of the function, and JL + 1 is the number of linearly independent
differentials, dv, of integrals of the first kind, which vanish in the zeros
of the function. When h + I vanishes the assigning of the poles of the
function, and of R — 1 of the zeros determines the other N—R + l, =p,
zeros ; in any case the assigning of the poles and of R — 1 of the zeros
determines the other N — R + l, =2) — (h+l), of the zeros. Denote the
poles by a1} ...,OLM and the assigned zeros by j31} ..., @R-i', then the remaining
zeros /3R, ..., ftN are determined by the congruences
0i ,« 0/(_i! « <*-i, a O-M, a „ er, a , , , .
Vi + . . . + Vi -Vi - ... - Vi - Z \.Vi - (ffi + A, Tit !+...+ IlpT{,p)
r-l
a being an arbitrary place. Now, let the form of the factorial function when
the poles are given be
where C\, ..., CR are arbitrary constants, and F1(ac), ..., FR(x) are linearly
independent ; then, when the zeros /3U ...,/3JC_1 are assigned, the function is a
constant multiple of the definite function
the zeros of this function, other than filt ..., fiR-1} are perfectly definite, and
are determined by the congruences put down. Let H denote the quantities
given by
Hi = 2 \.vc-' a + gi + /tjTt,! + ...... + hpTitp ;
r=l
take any places yl} ..., 7^+1, of assigned position, and take a place m and p
dependent places m1} ..., mp defined as in § 179, Chap. X., and consider the
function of #
if the function does not vanish identically, its zeros, x1} ..., xp, are (§ 179,
Chap. X.) given by the congruences denoted by
' m*+2
271] EXPRESSED IN FACTORS. 423
or, what is the same thing, by
now, from what has been said, it follows, comparing these congruences with
those connecting the poles and zeros of A (x), that if xl} ..., xh^ be taken at
7i, ..., 7/,+i, these congruences determine xh+y, ...,xp uniquely as the places
&K> •••> ftx- Thus the zeros of the theta function are the places 7lf ..., yh+i
together with the zeros, other than /31} ..., yS^_i, of the function A (x).
We suppose now M to be as great as p — 1, = r + p — 1 , say ; as in § 184,
p. 269, we take nly ..., ?ip_j to be the zeros of a (^-polynomial of which all the
zeros are of the second order, so that
, in _
is an odd half-period, equal to £Hgi s> say ; and we take the poles ar+1 ,...,«„ at
nlt ..., Wp_i. Further*, in this article, we denote
0 (v*> z + ins, ,.) e**'*'* by X (x, z\
so that (§ 175, Chap. X.) \(x, z) is also equal to e-^>+^'> 6(t>*. *; i«, I*').
The function X (x, z) must not be confounded with the function X (£, /A) of § 238.
Then in fact, denoting the arguments of the theta function by V, we
have the following important formula,
r h+l k
A (x) U X (a?, Zj) U X (x, 7j) II [X (x, C;)]*>
'
where ^. is a quantity independent of a;. In order to prove this it is
sufficient to shew (i) that the right-hand side represents a single-valued
function of x on the Riemann surface dissected by the 2p period loops,
(ii) that the right-hand side has no poles and has only the zeros of @(F),
and (iii) that the two sides of the equation have the same factor for every one
of the 2p period loops.
Now the function X (x, z) has no poles ; its zeros are the place z, and the
places ?ij, ..., np^. The places nl} ..., np^l occur on the right hand
(a) as poles, each once in A(#), each (R — 1) times in the product
r h+l
(p) as zeros, each r times in II X (x, a,-), h + l times in II X (as, 7;), and
.7 = 1 j = l
* For the introduction of the function X (x, z) see, beside the references given in chapter XIII.
(§ 250), also Clebsch u. Gordan, Abel. Functnen. pp. 251—256, and Riemann, Math. Werke (1870),
p. 134.
424 A THETA FUNCTION OF GENERAL ARGUMENT [271
k k
S \j times in II [A, (a, GJ)]^ ; thus these places occur as zeros, on the right
.7 = 1 .7 = 1
hand,
M- (p - 1) +h+ 1+ SX; - A = -AT- p + 1+ h + 1 - H,
times, that is, not at all.
Thus the expression on the right hand may be interpreted as a single-
valued function on the Riemann surface dissected by the 2p period loops —
for we have seen that the places nl} ..., np_l do not really occur, and the
multiplicity, at c/, in the value of such a factor as \\(x, c/)]A> is cancelled by
the assigned character of the factorial functions F(x) occurring in A(.r).
Nextly, the zeros of the denominator of the right-hand side, other than at
MI, ..., iip-i, are zeros of A (as), and the poles of A (#), other than nlt ..., np_l,
r
are zeros of the product II \(x, <Xj), so that the right-hand side remains
j=i
finite. The only remaining zeros of the right-hand side consist of 71, ..., yh+1
and the zeros of A(#) beside &, ..., /3^_j ; and we have proved that these are
the zeros of ® ( V). It remains then finally to examine the factors of the
two sides of the equation at the period loops. The factors of the left-hand
side at the i-ih period loops respectively of the first and second kind are
(see § 175, Chap. X.)
e - 27TJ (hi - K) and e ~ 2^* S (V - J«V) *>, i - 2iri ( Vt + K 4).
the factor of the right-hand side at the i-th period loop of the first kind is
e*, where
I
^r = - Vnrihi + ririsl + (h + 1) iriSi + irisl 2 \j — (R-l) iris- ;
;=i
k
now R = N - p + 1 + h + 1 = r + 2 X,- + h + 1 ; thus i/r = - 2Trih[ + 7m/, and
j=i
e* = e-*ri(hi-W)} or the factors of the two sides of the equation to be proved, at
the i-th period loop of the first kind, are the same. Since the factor of
X (x, z) at the t'-th period loop of the second kind is e^ where
i'~ + ^S{ + faiTi, 1 + • • • + s'pTi, P + %ri,
it follows that the factor of the right-hand side at the i-ih period loop of the
second kind is ex where
[T r a h + l Cf v k R~l T
v^'a;+ Svr^+SV?'- 2 v
j=i j=i j=i j=i
r h -i
- iri \r + h + 1 + l\j-R+l\ (T/ f + A-),
L .;=i
2
a!,|8.
272] EXPRESSED IN FACTORS. 425
now we have
a,, a a , a )i1(a
-vf -...-v-J -Vi
and
£ (si + S/TI, ,+ ...
thus
.7 = 1
further
_ x,a ,, n x a;, a. /T3 ^ •. x, a x, a ^ ~ x, a
0 = — Vi -\- \ll -f- 1 ) Vi — \l~i — i)Vi -f- TVi T ^ "-jfj ^
hence
A+l 3.^. JB-l a.) /3. r x> a
or
/* r h+l h B-l
S(AM-K')^i+^--flr,- + i*,-+ 2t;?*>+ S vf v> + S \X ' - ^ v?'*,
/* = ! ./ = ! ? = 1 J = l ;' = !
and thence the identity of the factors taken by the two sides of the equation
to be proved, at the t'-th period loop of the second kind, is manifest.
And before passing on it is necessary to point out that if the functions
X (x z}
\ (x, z) be everywhere replaced by r— , and A (x) be replaced by i/r A (x},
^r being any quantity whatever, the value of the right-hand side of the
equation is unaltered. For there are R factors A, (x, z} occurring in the
numerator of the right-hand side of the equation beside A (x), and R — 1
factors \(x, z) occurring in the denominator of the right-hand side of the
equation. In particular ty may be a function of x.
272. We can now state the following result: Let a, «n ..., «r be any
assigned places; let TO,, n.2, ..., ?tj>_1 be the zeros of a ^-polynomial, or of
a differential, dv, of the first kind, of which all the zeros are of the second
order, and
>u,,,m n,,»i, n,,-i, mt>-i , ,
Vi -V{ -...-Vi =$(si + si'T;il+...+Sp/Ti,p), (l = 1, 2, ...,})),
111, MI,, ..., mp being such places as in § 179, Chap. X.; let li+l be the
number of linearly independent differentials, dv, which vanish in the zeros of
a factorial function of the primary system having cti, ..., or,., nlt ..., np_1 as
42(j STATEMENT OF THE RESULT. [272
poles, or the number of differentials dV, of everywhere finite factorial
integrals of the associated system, which vanish in the places nlt ..., np_1}
k
a,, ..., a,.; let yl, ..., yh+i be any assigned places; denote r + 2 \j + h+l
j=i
by R, and let* xl} ..., XR be any assigned places; let the general factorial
function of the primary system having a1} ..., a.,, n1} ..., »ip_x as poles be
0^(0,)+ + CRFR(x),
wherein Glt ..., GB are constants, and let
(X), ,FR(xl] \^(xl),...,^(xK))
,FH(xB}
where ty (x) denotes any function whatever ; let
7? y 7i 4- 1 Z-
Tr £ x.,a £ a., a "JT1 y.,a £ c., a
U{= 2tv - 2 Vi>' - 2 v - 2 \jvf ,
j = l j=l j=l ./ = !
which is independent of a, and let the row of^> quantities
i > x ...... p - p it p
be denotedf by g-^8 + r(h-^8')j then if, modifying the definition of
X (x, z\ we put
we have
A. (a- r r"» R ( r h+l k
= t,j-i,i' ,2V" n n \(^ «,) n x (^ 7j) n [x^,, Cj)
nri x (xit Xj} i=l u=1 •;'=1
i<j
wherein (7 is a quantity independent of xlt ..., XR, which may depend on
Cj, ..., Cfc, «!, ..., «,., yl} ..., 7A+1.
273. The formula just obtained is of great generality; before passing
to examples of its application it is desirable to explain the origin of a certain
function which may be used in place of the unassigned function ty (as).
We have (§ 187, p. 274), in the notation of § 272,
if the zeros of the rational function of x, (x1 — x)j(x' — 2), be denoted by
* These replace the xl, ft, ... , ^..j of § 271.
t So that V= U - (g - $g) - r (h - ^*').
273] THE SCHOTTKY-KLE1N PRIME FORM. 427
x, xl, ..., c£H_i, n being the number of sheets of the fundamental Riemann
surface, and the poles of the same function be denoted by z, zlt ..., zn_^ we
have, by Abel's theorem,
= 10 ^'-^_lr_
now let the places x, z' approach respectively indefinitely near to the places
x, z which, firstly, we suppose to be finite places and not branch places ; then
the right-hand side of the equation just obtained becomes
log \_-(x=zY " X(cc)X(z)
where
p x „ v
\ i /v» i „ ^> (—\ f 1 (~\ \ 7~i ~V / \ ^*
•"^ \ *™ ) — "^ ^^ / \ "9" •* ^x v' / • -*^ ^/ i -^*- \ -^ / ^~ - /
Z) denoting a differentiation, and a denoting an arbitrary place ; but we have
(Chap. X. § 175)
thus, on the whole, when the square roots are properly interpreted, we
obtain
/ — — aTz'
\un.X'=x , •,__, \/ — (x' — x) (z' — z) e x>z = — *'—"—-
"V A/V/\V/\
\i \ i np i \ i^i
» -*v V / ^ V /
When the places «, ^ are finite branch places we obtain a similar result.
Denote the infinitesimals at these places by t, t1} and, when x, z' are near to
as, z, respectively, suppose of = x + tw+1, z =z + ^1+1 ; then from the equation
given by Abel's theorem we obtain, if 7 denote an arbitrary place,
w r x' z1 i "j:1 v' / wi r r- / i »-! -^ ,-
S n£; - log d + s nj;,; + 2 n^;: - log d + 2 nj^
=l L J r=w+l r=l L J r=w,+l
where X (#), X (2) are of the same form as before, save that the differentia
tions Dvi' , Dvi , are to be performed in regard to the infinitesimals t, ^.
If the limit of the first member of this equation, as x, z' respectively
approach to x, z, be denoted by Z, we therefore have
- IT* ' " H (vx> z -4- ill -\
**U +g= =(x-z)e^ (ii)
x) X(z)
428 THE SCHOTTKY-KLE1N PRIME FORM. [273
The equations (i), (ii) are very noticeable ; there is no position of x for
which the expression ® (vx' z + J Hg) g') . e^18**' z/vX (x) X (z) is infinite, and there
is only one position of so, namely when x is at z, for which the expression
vanishes; for (§ 188, p. 281) the expression *J X (x) vanishes, to the first
order, only when x is at one of the places nlt ..., np^, and ® (vx> z + ^£ls,s')
vanishes only when x is at one of the places z, nlt ..., np^ ; there is no
position of x for which \/X (x} is infinite. Putting
T(x ._®(
we have further •GTI (x, z} = — TSI (z, x}, and if t denote the infinitesimal near
to z, we have, as x approaches to z, limita;=z [CTZ (x, z)jt\ = 1. For every
position of a; and z on the dissected Riemann surface •sr1 (x, z) has a perfectly
determinate value, save for an ambiguity of sign, and, as follows from
the equations (i), (ii), this value is independent of the characteristic
(H iO-
There are various ways of dealing with the ambiguity in sign of the
function tal (x, z). For instance, let (f> (x) be any ^-polynomial vanishing in
an arbitrary place m, and in the places A1} ..., A.2p_3 (cf. § 244, Chap. XIII.),
and let Z (x) be that polynomial of the third degree in the p fundamental
linearly independent ^-polynomials which vanishes to the second order in
A1)...,A2p^3 and in the places ml} ..., mp. Further let 3?(x) be that
^-polynomial which vanishes to the second order in the places nj, ..., ?ip_!.
Then we have shewn (§ 244) that the ratio \/Z (x)/(f> (x) \/<I> (x), save for an
initial determination of sign for an arbitrary position of x, is single-valued on
the dissected Riemann surface ; hence instead of the function CTJ (x, z) we
may use the function
E (x z} =
which has the properties ; (i) on the dissected Riemann surface it is a single-
valued function of x and of z, (ii) El (x, z) = — E1 (z, x}, (iii) as a function of x
it has, beside the fixed zeros m1, ..., mp, only the zero given by x = z, and it
has no infinities beside the fixed infinity given by x = m, where it is infinite
to the first order. At the r-th period loops respectively of the first and
second kind it has the factors
I g- *»<<«»? *+ tort)
But there can be no doubt, in view of the considerations advanced in
chapter XII. of the present volume, as to the way in which the ambiguity of
the sign of •srl (x, z} ought to be dealt with. Suppose that the Riemann
surface now under consideration has arisen from the consideration of the
273] THE SCHOTTKY-KLEIN PRIME FORM. 429
functions there considered (§ 227) which are unaltered by the linear substitu
tions of the group. Let the places in the region S of the f plane which
correspond to the places x, z, x ', z' of the Riemann surface be denoted by
£> £• £ . £"• Then by comparing the equation obtained in chapter XII. (§ 234),
with the equation here obtained,
z
and noticing that X (x), -^ agree in being differential coefficients of an
a£
integral of the first kind, which vanish to the second order at w,, ..., np^l,
we deduce the equation
, . I / dt dt^
<«.*>/V3 "3?
f "
now we have shewn that ro-(£, £) is a single-valued function of f and £; and
any one of the infinite number of values of £, which correspond to any value
of #, has a continuous and definite variation as x varies in a continuous way;
hence it is possible, dividing «TJ (x, z) by the factor A / -^ . -^ , which by
V Okc^ ft'^
itself is of ambiguous sign, to destroy the original ambiguity while retaining
the essential character of the function CTJ (x, z). The modified function is
infinitely many-valued, but each branch is separable from the others by a
conformal representation. Thus the question of the ambiguity in the sign of
CTI (x, z) is subsequent to the enquiry as to the function £ which will conform
ably represent the Riemann surface upon a single f plane in a manner
analogous to that contemplated in chapter XII. §§ 227, 230*.
In what follows however we do not need to enter into the question of the
sign of vrl (x, z}. It has been shewn in the preceding article that the final
formula obtained is independent of the form taken for the function there
denoted by ty (x). It is therefore permissible, for any position of x, to take
for it the expression VZ (x), with any assigned sign, without attempting to
give a law for the continuous variation of this expression. The advantage is
in the greater simplicity of nyl (x, z) ; for example, when x is at any one
* Klein has proposed to deal with the function •srl (x, z) by means of homogeneous variables.
The reader may compare Math. Annal. xxxvi. (1890) p. 12, and Eitter, Math. Annal. XLIV. (1894)
pp. 274 — 284. In the theory of automorphic functions the necessity for homogeneous variables
is well established. Cf. § 279 of the present chapter. For the theory of the function •&1 (x, z) in
the hyperelliptic case see Klein, and Burkhardt, Math. Annul, xxxn. (1888).
430
THE GENERAL FORMULA FOR ROOT-FUNCTIOXS.
[273
of the places nlt ..., np_lt the function \(as, z), as denned in § 271, vanishes
independently of z ; but this is not the case for TS^ (x, z).
Ex. i. Prove that
c)
*' z=\oe i i
«•« B »,(*,«)»,(* a)'
Ex. ii. Prove that any rational function of which the poles are at at, ... , a.,, and the
zeros at /3j , . . . , /Sj,, can be put into the form
where
**!(#, a^ ...... or^a?, a*)
j, ... , X;, are constants, and a is a fixed place.
Tn what follows, as no misunderstanding is to be apprehended, we shall
omit the suffix in the expression ^l (x, z}, and denote it by •OT (as, z). The
function -a (£ £) of chapter XII. does not recur in this chapter.
274. As an application of the formula of § 272 we take the case of the
root form \/X<3> (x)j® (as)*/ X (x), where Z(3) (x) is a cubic polynomial of the
differential coefficients of the integrals of the first kind, having 3 (p - 1) zeros,
each of the second order (cf. § 244, Chap. XIII.). Then the poles a1} ..., a,, are
the 2p — 2 zeros of any given polynomial <I> (x), which is linear in the
differential coefficients of integrals of the first kind. Thus r = 2p — 2,
A + 1 = 0, R, =r + h+I + '2\j =2p -2 + 0 + 0 = 2p -2; U=P^ ^>"y, and,
i i
taking for the function i/r (x), the expression V X (x), the formula becomes
"
2p-2 2p-S
n n
i, j=l, 2, ...,
n n
n
Herein 4>(«) is a given polynomial with zeros at Oj, ..., a»p-2, and the forms
v JTj («), ...,
_^(x) are any set of linearly independent forms, derived
as in § 245, Chap. XIII., and having (—g1,..., — h-l,..., — hp) for characteristic.
From this formula* that of § 250, Chap. XIII. is immediately obtainable.
The result is clearly capable of extension to the case of a function
* Cf. Weber, Theorie der AbeVschen Functionen vom Geschlecht 3, Berlin, 1876, § 24, p. 156;
Noether, Math. Annul, xxvm. (1887), p. 367; Klein, Math. Annal xxxvi. (1890), p. 40. For the
introduction of ^-polynomials as homogeneous variables cf. §§ 110—114, Chap. VI. of the present
volume. See also Stahl, Crelle, cxi. (1893), p. 106; Pick, Math. Annal. xxix. " Zur Theorie der
AbeFschen Functionen."
276]
THE GENERAL FORMULA FOR RATIONAL FUNCTIONS.
431
275. A general application of the formula of § 272 to the case of rational
functions may be made by taking al} ..., a,, to be any places whatever, r
being greater than p-l. Then A + 1=0 and R = r\ and if the general
rational function with poles in a,, ..., ar, nl} ..., np,l be
A-Fi (x) + + A^F^ (x) + A,,
where Aly ..^Ar are constants, and we take for the function ^ (x) the
expression V 'X (x\ and modify the constant C which depends in general upon
«,, ..., <xr, we obtain the result (cf. § 175, Chap. X.)
, ..
II
II
f, O;)
^T(xl)...X(xr}X(a,} ...X (
276. This formula includes many particular cases*. We proceed to
obtain a more special formula, deduced directly from the result of § 272.
Let «!, .... ar = nll ..., np_lf Then the everywhere finite factorial integrals
of the associated system are the ordinary integrals of the first kind,
and the number, h + 1, of dV which vanish in the places a,, ..., aj,
nl} ..., ?i_p_1> that is, which vanish to the second order in the places
i, is 1. The number ^, =
general function having the poles
1, =
, =p. The
...,?iVi mf(»)»^(te)/Z(ai)t where
X (x) is the expression employed in § 273, and <& (x) denotes the differential
coefficient of the general integral of the first kind. Further
a- 2 vni'a-
i
7 being an arbitrary place. Hence
-8- rs =
and
eirw'df+J.+ i
is equal (§175, Chap. X.) to
= say,
.', @ ( 7+
since ss is an odd integer. Therefore taking for the function
expression VZ (ar), X (x, z) is CT (x, z), and
A (f T \ —
LA \^i, . . . , O/y.^ —
(x} the
. xxxvi. p. 38.
432 THE GENERAL FORMULA FOR RATIONAL FUNCTIONS. [276
where <I> (a;), . . . , 4>p (x) denote dv*' "fdt, ..., dv% */dt. Thus on the whole
-^l- n * *
..
n n
where G is a quantity which, beside the fixed constants of the surface, depends
only on the place 7. Let us denote the expression
which clearly has no zeros or poles, by p (#;) ; then we proceed to shew that
in fact G = Ap, (7), where A is a quantity depending only on the fixed
constants of the surface, so that we shall have the formula
™' v 0 ( V) =
n n •nr (xi, Xj} /i (7)
i<j
where
p
i
In this formula 7 only occurs in the factors
_
—
herein the factor JT (7) occurs once in the denominator of each of CT(^-, 7),
and p times as a denominator in /j, (7) ; thus this factor does not occur at all.
In determining the factors of M*, as a function of 7, it will therefore be suffi
cient to omit this factor. Thus the factor of ^ at the t'-th period loop of the
first kind is enis'(p~P~l) or e™' . At the t'-th period loop of the second kind the
factor of ®(vx'z + %(ls,S') e™'1'*' * is e-2l">f>e+K. •)-"*«, and therefore the
factor of ^ is
e - iris, - 2*i (vy' XP + vn> ' *4 ...... + f71"-1 ' Xp'1 + IT,. ,).
Consider now the expression
at the i-th period loop of the first kind, this function, regarded as depending
upon 7, has the factor e™' ; at the i-ih period loop of the second kind it has
the factor
TJH
277] THE FORMULA FOR THE HYPERELLIPTIC CASE. 433
but since
iti (Si + Ti,ls\+ ...... + Ti.ps'y)
it follows that
is equal to
- irisi
thus, changing 7 into x, we have proved that the function of x
has the same factors at the period loop as the function, of x, given by
w 0», #1) ...... «• (#, arp)//* (a;) ;
it is clear that these functions have the same zeros, and no poles.
Hence the formula set down is completely established*.
277. We pass now to the particular case of the formula of § 272 which
arises when the fundamental Riemann surface is hyperelliptic, and associated
with the equation
if = 4*
Then the places n1} ..., np^ are branch places. We suppose also that p + 1
of the places alt ...,ar are branch places, say the place for which x = dl, ...,
c£M+1) and that //. + 1 of the places xly ...,xr are branch places, say those
at which # = b1} ..., b^+1. It is assumed that the branch places n1} ...,
WP_,, dl} ..., c^+i, b1} ..., 6^+1 are different from one another. We put
r — (/j,+ l) = v, then the determinant of the functions Ft (xj\ (§ 272),
regarded as a function of asl, is a rational function with poles in nl} ... , Wp_1}
flj, ..., a.v, d1} ..., c?M+1 and zero in ar2, ..., «„, 6X, ..., 6M+1. Provided v is not
less than //,, such a function is of the form
-ttl^-fo-Wp.^-^...^^^
(^-^."(afi-Wp-iX^-^ — C^-^+iX^-ai)--- («i-«r)
where the degrees of (a?,, !)„_!_„, (a?,, 1),_1+M are determined by the condition
that the function is not to become infinite when a^ is infinite. When v = //,,
the terms (a^, l)^.^ are to be absent. When i> < //,, the conditions assigned
do not determine the function ; we shall suppose v 5 /A. The 2y — 1 ratios
of the coefficients in the numerator are to be determined by the conditions
that the numerator vanishes in x2, ...,#„ and in the places conjugatef
* See the references given in the note *, § 274, and in particular Klein, Math. Annul, xxxvi.
p. 39.
t The place conjugate to (x, y) is (x, -y)
B- 28
434
DEFINITION OF THE HYPERELLIPTIC THETA FUNCTION
[277
to «!, ..., «„. Hence, save for a factor independent of x1} the determinant
of the functions F{ (xj) is given by
wherein i/r (as) = (x-n1)...(x- n____) (x-d^)...(x- d^) (x-b^...(x- 6M+1),
0 (a?) = y^l^r (x), and the determinant has 2v rows and columns ; denoting
this determinant by D^ +, the determinant of the functions Fi (xj) (§ 272) is
therefore equal to
* 1 /(xi-b1)...(xi-
' * i=i (a?i-o1)...(a?<-a,)V(a;i-n1)...(a!f-v5 V («» - dj. ..(xt-
-bl.+l)
Hence, from § 272, taking ^(x) = ^(x -n,) ...(x-n^), so that «• (a;, *) will
denote
we have
where C is independent of #_, ..., xv.
Now, if b, d be any two branch places, and a an assigned branch place,
and hence, if
r, 6)
b,a
where ft, ..., /Op', S1} ..., 8/ are integers, we have (§ 175, Chap. X.)
277] BY THE FUNCTION ta (x, z) AND ALGEBRAICAL FUNCTIONS. 435
where A is independent of x. Thus the expression
„• (8Hn & a *r (x, d) /x-b
vr(x,b) V x-d'
which clearly has no poles or zeros, is such that its factors at the period loops
are all + 1. The square of this function is therefore a constant, and the
expression itself is a constant.
Therefore if
M+l di l.
« «< ' » f(<r< 4 a-i'Tt , i + ...... + <rp'Ti>p),
where o-,, ...,<rp are integers, it follows that the function
e-Ki<r'(vx"a> + ...... +t*»M jj Mjj^0t, <£;) /Xj - bj
i = l j=l -57 (#i, bj) V «fc — (^
is independent of a^, ..., a?,,. Further
@ (u - \<r - \-rcr ; \s, \s} = Be™'* ®[u; %(S-<T),I (s - a-')]
by § 175, Chap. X. Thus on the whole we have
eejji «•»••«; K'-'XW-
, , t-« ^ a
•<j f=ij=i j<j
where C is independent of ^, ..., a?,. Hence we can infer that C is in fact
independent also of a, ,...,«„. For when the sets xlt ...,#„, alt ...,«„ are
interchanged, ^^ is multiplied by (-)"2^-M = (_ I}M and> since CT (^ ^
= - -sr (^ «), this is also the factor by which the whole right-hand side is
multiplied. The theta function on the left-hand side is also multiplied
by + 1. Thus the square of the ratio of the right-hand side to the theta
function on the left is unaltered by the interchange of the set xlt ..., xv with
the set «,, ..., «„. Thus C2 is independent of ^, ..., Xv and unaltered when
a?,, .... av are changed into a,, ...,«„. Hence C is an absolute constant.
It follows that the characteristic *(* - er), W - *\ and the theta
functions, are even or odd according as ^ is even or odd.
In the notation of § 200, Chap. XL, the half-periods \^, are given by
hence, if the half-periods given by
be denoted by ^ft, the half-periods associated with the characteristic
f(*-*)i $(s - o-') are congruent to expressions given by
28—2
436 RULE FOR THE THETA-CHARACTERISTIC [277
while -fy, which is of degree p + 1 + 2/i, is. equal to
(an- O ...(a?- n^) (x - &0 ...(a?- 6M+1) (a; - dj ... (a; - d^+1);
by means of the formula (§ 201, Chap. XI.)
the half-periods associated with the characteristic ^(s — a), ^(s' — <r') can be
reduced to be congruent to expressions denoted by
where elt ..., ep-2li+1 are given by
also, in taking all possible odd half-periods £ng)S<,all possible sets of p — 1
of the branch places will arise for the set nly ..., np^. Hence it follows that
the formula obtained includes as many results as there are ways of resolving
(x, l)ap+2 into two factors ^+1_2F, T/r^+i+2M> °f orders p + 1 — 2/A, p + 1 + 2/4,
and (§ 201) that all possible half-integer characteristics arise, each associated
with such a resolution. We have in fact, corresponding to fj, = 0, 1, 2, ...,
E (^—~ — ) , a number of resolutions given by
It has been shewn (§ 273) that the expression ta (x, z) may be derived,
by proceeding to a limit, from the integral II Jt J. Hence the formula that
has been obtained furnishes a definition of the theta function in terms
of the algebraic functions and their integrals, and has been considered from
this point of view by Klein, to whom it is due. After the investigation
given above it is sufficient to refer* the reader, for further development, to
Klein, Math. Annal. xxxn. (1888), p. 351, and to the papers there quoted.
Ex. i. Prove that the function 6 [u ; i (s - <r), i (*' - o-')] vanishes to the /*th order for
zero values of the arguments.
Ex. ii. In the notation of § 200, Chap. XL, prove, from the result here obtained, that
each of the sums
r+2 Ci,a a,, a <r+4 cita a,,a , 4r+3 cifa
2 v* , v ' + 2 vl , v } + 2 vl
represents an odd half- period ; here c< is any one of the places c,clt ... , cp, a» is any one of
the places «15 ... , ap, a,- is any one of the places a1} ... , ap, and r is an arbitrary integer
* See also Brill, Grelle, LXV. (1866), p. 273; and the paper of Bolza, American Journal, vol.
xvii., referred to § 221, note, where Klein's formula is fundamental.
By means of the rule investigated on page 298, of the present volume, the characteristic
£ (s - ff), J (s' - ff') can be immediately calculated from the formula here (p. 436) given for it. Cf.,
also, Burkhardt, Math. Annal. xxxii., p. 426; Thompson, American Journal, xv. (1893), p. 91.
278] WHEN THE ALGEBRAICAL CHARACTERISTIC IS GIVEN. 437
whose least value is zero, and whose greatest value is given by the condition that i cannot
be greater than p + 1. Prove also that each of the sums
4r+l 4r 4r4-2 4r-M
2 /'•'", S/*°, *«*•+*%**, va*'a+ l
1=1 i=l i=l t=l
represents an even half-period. For a more general result cf. the examples of § 303 (Chap.
XVII.).
Ex. iii. By taking i/=p + l,/x = 0, and the places b, d so that ^Qit s, = iP>d) finally
putting nv, ... , np_l, b, d for a^ ... , ap, aj, + 1, obtain, from the formula, the result
(x, q) s-q P (x-xi
izr (.£, #;) zzr (2. Ov)
— — '
i a;, z i , izr .£, #; zzr 2. Ov . , , . .
where n replaces log — j-2 — K — p — L' J, (^) = (^ - a) ... (x — «„), and the branch places
xi'ai TZ (X, tti) •& (z, Xi) '
a, a, , ... , ap are, as in § 203, Chap. XI., such that the theta function in the numerator of
the left-hand side vanishes as a function of x at the places £1? ..., gp, conjugate to
x± , . . . , xp ; and verify the result d priori. By the substitution
xi,ai=e
this formula can be further simplified. Deduce the results
x.z x,* =1 e(^>a-pa?"a'- ...... -tr^p»a
n - °g
9 dx
where w = 'y-r"ai-|- ...... + vx"' a", Zi(u) = ^-\oge(u), and -^ ,... are as in § 123, Chap. VII.
These results have already been given (Chap. X.).
278. It is immediately proved, by the formula (§ 187)
_
-
that the general expression of a factorial function given in § 254 can be
written in the form
n [®(^.-i + ^n, ^e™'"*'*^ n
1 L J i L
And, by the use of the expression TS (x, z), this may be put into the form
_2Tri£()l. + H.)vx'y £ M [ "I"1 * f ~|-\
t f *' t [ *> «* J t [CT (*• ^ J
438 CONNECTION OF THEORY OF FACTORIAL FUNCTIONS [278
Ex. i. In the hyperelliptic case associated with an equation of the form
y* = (x,l)w + s,
if x denote the place conjugate to the place x, it follows from the formula of § 273 that
in*'2
•si (x, z} = (x — z)e2 x,z,
unless x or z is a branch place.
Ex. ii. In the hyperelliptic case, if £, &15 ...,£„ denote branch places, and
and the equation associated with the surface be yz = f(x\ where f(x) = <£(#) ty(x\ and if
we take places x, xv, ... , xp, z, zlt ... , zp, such that
<»*' + ...... + „?'*>=$*, %•*> + ...... + i>J"*"Stf*, (t=l,2,...fp),
then it is easily seen that the rational function having x,x^ ..., xp as zeros and Z,ZI}...,ZP
as poles, can be put into the form [y1^ (x)+y(f> (•*')] -^I» (*) + «<£ (*01 wb-ere ^> / are the
variables and s is the value of y1 at the place z. Hence prove, by Abel's theorem, that
Ex. iii. Suppose now that a, a^ ..., «p are the branch places used in chapter XL
(§ 200), so that
...... +u
x>z
and suppose further that |Q,=£(s + rs')> is an even half-period such that
and
then deduce that
_
The results of examples i, ii, iii are given by Klein.
Ex. iv. Prove that, if z, f, cx , . . . , cp be arbitrary places, and yv , . . . , yp be such that
the places £, yls ... , yp are coresidual with the places z, ct, ... , cp, then
p -&(x,z)w (f, z)
hence deduce, by means of the result given in Ex. iv., page 174, that
where a is an arbitrary place.
279. The theory of the present chapter may be considered from another
point of view. We have already seen, in chapter XII., that the theory of
rational functions and their integrals may be derived with a fundamental
surface consisting of a portion of a single plane bounded by circles, and the
279]
WITH THEORY OF AUTOMORPHIC FUNCTIONS.
439
change of independent variables involved justified itself by suggesting an
important function, CT (f, 7). We explain now*, as briefly as possible, a more
general case, in which the singular points, c1} ..., Ck, of this chapter, are
brought into evidence.
Suppose that a function £ exists whereby the Riemann surface, dissected
as in § 253, can be conformally represented upon the inside of a closed
curvilinear polygon, in the plane of f, whose sides are arcs of circles^; to the
four sides, (a;), (a/), (&;), (&/), of a period-pair-loop are to correspond four sides
of the polygon, to the two sides of a cut (7) are to correspond two sides of
the polygon ; the polygon will therefore have 2 (2p -f k) sides.
Fig. 11.
Then it is easily seen that if C be the value of £ at the angular point C of
the polygon, which corresponds to one of the singular points c1} ..., Ck on the
Riemann surface, and D be the value of £ at the other intersection]: of the
circular arcs which contain the sides of the polygon meeting in C, we can
pass from one of these sides to the other by a substitution of the form
where ITTJI is the angle G of the polygon, (I being supposed an integer other
than zero) ; as we pass from a point £ of one of these sides to the corresponding
point of the other side, the argument of the function [(£- C)/(£- D)]* increases
by 2?r ; if therefore t be the infinitesimal at the corresponding singular point on
the Riemann surface, we may write, for small values of t, (f- C)/(£-D) = tf ,
i i_
so that £- C = t' (C - D) (1 - *)-!. Further if £ £ be corresponding points
* Klein, Math. Annal. xxi. (1883), "Neue Beitrage zur Kiemann'schen Functionentheorie " ;
Ritter, Math. Annal. XLI. (1893), p. 4; Ritter, Math. Annal. XLIV. (1894), p. 342.
t See Forsyth, Theory of Functions, chapter XXII., Poincare, Acta Math, vol's. i.— v. We may
suppose that the polygon is such as gives rise to single-valued automorphic functions.
J Supposed to be outside the curvilinear polygon.
440 A FACTORIAL FUNCTION MAY BE REGARDED [279
011 the sides of the polygon which meet in C, we have for small values of t,
" "*' = * " '*'
d£ = (C _ D) tl dt, d? = (C-D) t e V ' dt, or
V V
I
ultimately, the factor omitted being a power series in tl or (£ — C)/(£— D),
whose first term is unity.
We shall suppose now that the numbers Xx, ..., \k of this chapter are
given by X^ = — mt/li, where mi, li are positive integers. Then a function
whose behaviour near a is that of an expression of the form t~* </>, will, near
Gi, behave like (£ — Ci)mi<f>, that is, will vanish a certain integral number of
times. Further, for a purpose to be afterwards explained, we shall adjoin to
the k singular points cl5 . .., Ck, m others, ely ..., em, for each of which the
numbers X are the same and equal to — e, so that, if t be the infinitesimal
at any one of the places el} ..., em, the factorial functions considered behave
like £'<£ at this place. These additional singular points, like the old, are
supposed to be taken out from the surface by means of cuts (ej), . . . , (em) ;
and it is supposed that the corresponding curves in the curvilinear polygon
of the £-plane are also cuts passing to the interior of the polygon, as in the
figure, so that at the point E^ of the £-plane which corresponds to the place el
of the Riemann surface, £ is of the form ^ = E^-\- t<j>, where <f> is finite and not
zero for small values of t, t being the infinitesimal at el,
Factorial functions having these new singular points as well as the
original singular points will be denoted by a bar placed over the top.
Let dv denote the differential of an ordinary Riemann integral of the
first kind which has p — I zeros of the second order, at the places
MI, ..., %_j. Consider the function
where a, c are arbitrary places, and p is determined so that Z2 is not
infinite at the place c, or
this function is nowhere infinite on the Riemann surface ; it vanishes to the
first order only at £ = oo ; for each of the cuts (e^, . . . , (em) it has a factor
nip 1
em ; at a singular point c; it is expressible as a power series in t1 , or
(?— ty/(£-D), whose first term is unity. The values of Z2 at the two sides
of a period loop are such that Z2'/Z2 = Vc^/c^' ; but since these two sides
correspond, on the £-plane, to arcs of circles which can be transformed into
one another by a substitution of the form £' = (a£ + /3)/(7£ + 8), wherein we
suppose aS - fiy = 1, it follows that Z2'/Z2 = y^+8. If then we also introduce
279] AS AN AUTOMORPHIC FORM. 441
the function Zlt = £Z.2> we have for the two sides of a period loop, equations of
the form
Za' = j
Consider now a function
where K is a factorial function with the k + m singular points, and R=2me/p.
i
At a singular point C;, or (7{, its behaviour is that of a power series in r or
(£_(?)/(£_£), multiplied by (?-Ci)mi; at a singular point a, or Eit its
behaviour is that of a power series in the infinitesimal t multiplied by
or unity ; at a period loop it is multiplied by a factor of the form p, (y%+ &)~R,
where p, is the factor of K. The function has therefore the properties of
functions expressible by series of the form*
wherein the notation is, that ft83 (&t £+&)/(?< £+$0 is one °f the finite
number of substitutions whereby the sides of the curvilinear polygon are
related in pairs and R(£i) is a rational function of &. The equation
connecting the values /', /, of the function /, at the two sides of a period
loop, may be put into the form
and we may regard Z.2 f, or K, as a homogeneous form in the variables
Zlt Z2, of dimension R.
The difference between the number of zeros and poles of such a factorial
function K is (§ 254)
adding the proper corrections for the zeros of the automorphic form K at
the angular points Gl, ..., Ck (Forsyth, Theory of Functions, p. 645) we have,
for the excess of the number of zeros of the automorphic form over the
number of poles
2 X + 2 = - 2 - 2 + k + m + 1 - 2 + m + 1
*
where q = k + m + 1, 2 - = 2 7- + m + 1.
/^ ti
We may identify this result with a known formula for automorphic
* Forsyth, Theory of Functions, p. 642. The quantity R is, in Forsyth, taken equal to - '2m.
442 HOMOGENEOUS VARIABLES ON THE RIEMANN SURFACE. [279
functions Forsyth, Theory of Functions, p. 648 ; if in the formula
m ( n — 1 — 2 - ) , there given, we substitute, by the formula of p. 608, § 293,
V f^/
n = 2JV— 1 + q, we obtain m (ZN— 2 + q — 2 - ] ; for each of the angular
V A1/ J
points G!, ..., Ck is a cycle by itself, each of the points E1} ..., Em is a cycle
by itself, and the remaining angular points together constitute one cycle
(cf. Forsyth, p. 596) ; the sum of the angles at the first k cycles is 2?rS T- ,
k
the sum of the angles at the second m cycles is 2?rm, the sum of the angles
at the other cycle is 2?r*.
There is a way in which the adjoint system of singular points elt ..., em
may be eliminated from consideration. Imagine a continuously varying
quantity, xZ) which is zero to the first order at et, ..., em and is never infinite,
and put xl = xxz ; the expression Kx2~e may then be regarded as a homo
geneous form in xlt #2 on the Riemann surface, without singular points at
el , ..., em\ and instead of the function Zz we may introduce the form
_j>_
£2 = Z^x2 Zm , which is then without factor for the cuts (ej), . . . , (em), or, as we
may say, is unbranched at the places e1; ..., em ; and may also put ^= ^2.
Thus, (i), a factorial function, considered on the £-plane, is a homogeneous
automorphic form, (ii), introducing homogeneous variables on the Riemann
surface, the consideration of factorial functions may be replaced by the con
sideration of homogeneous factorial forms.
Ex. Shew that the form
1 f-fX, a 1 /r,x, a . r,x, a \ , „ x x, a z, c
- n' — (n + +n j+sXj.-v. v.
n, \ »« /•/ \ z, c m\ eltc ' em, c' , , t'-7 t j
P(x,z)=x2 f(z)e *J ,
where a, c are arbitrary places and X£i y are constants, is unbranched at e^ ... , em, that it
has no poles, and vanishes only at the place z. Here f(z) is to be chosen so that, when x
approaches 2, the ratio of P (x, z) to the infinitesimal at z is unity. At the t-th period
loop of the second kind the function has a factor ( — )•* where
If n • i ^7r* / / \ "ifl i C,, C . . 6m, CN . _ N z, c
M=2mr+—(q't-q)-—(vt1' + +^ ) + ^ _Xi>y v' ° rit t ,
q'2 — q denoting the number of circuits, made in passing from one side of the period loop to
the other, of x2 about #2=0 other than those for which x encloses places elt ... , em, and r
denoting the number of circuits t of x about z.
* The formula is given by Ritter, Math. Annal. XLIV. p. 360 (at the top), the quantity there
denoted by q being here - \ p. We do not enter into the conditions that the automorphic form
be single-valued.
t The reader will compare the formula given by Ritter, Math. Annal. XLIV. p. 291. It may be
desirable to call attention to the fact that the notation <r + 1, <r' + 1, as here used, does not coincide
with that used by Ritter. The quantities denoted by him by a, a1 may, in a sense, be said to
correspond respectively to those denoted here, for the factorial system including the singular
points elt ..., em, by <r' + l and w'.
281]
CHAPTER XV.
RELATIONS CONNECTING PRODUCTS OF THETA FUNCTIONS — INTRODUCTORY.
280. As preparatory to the general theory of multiply-periodic functions
of several variables, and on account of the intrinsic interest of the subject, the
study of the algebraic relations connecting the theta functions is of great
importance. The multiplicity and the complexity of these relations render
any adequate account of them a matter of difficulty ; in this volume the plan
adopted is as follows : — In the present chapter are given some preliminary
general results frequently used in what follows, with some examples of their
application. The following Chapter (XVI.) gives an account of a general
method of obtaining theta relations by actual multiplication of the infinite
series. In Chapter XVII. a remarkable theory of groups of half-integer
characteristics, elaborated by Frobenius, is explained, with some of the theta
relations that result ; from these the reader will perceive that the theory is of
great generality and capable of enormous development. References to the
literature, which deals mostly with the case of half-integer characteristics, are
given at the beginning of Chapter XVII.
281. Let <j>(ul} ...,Up) be a single-valued function of p independent
variables wlf ...,vp, such that, if a1} ...,ap be a set of finite values for
MJ, ..., up respectively, the value of <j> (u^, ..., up), for any set of finite values
of Wj, ...,up, is expressible by a converging series of ascending integral
positive powers of u^ - a1} u^ — a2, . . . , up — ap. Such a function is an integral
analytical function. Suppose further that <f)(u1} ..., up) has for each of its
arguments, independently of the others, the period unity, so that if m be any
integer, we have, for a = 1, 2, . .. , p, the equation
Then* the function <£(?*!, ..., up) can be expressed by an infinite series of
the form
S ...... 2 Ani .....
W,= -oo wp=-oo
* For the nomenclature and another proof of the theorem, see Weierstrass, Abhandlungen
aiis dcr Functionenlehre (Berlin, 1886), p. 159, etc.
444 PROOF OF THE FUNDAMENTAL [281
wherein n1} ...,np are integers, each taking, independently of the others, all
positive and negative values, and AHj> ..., np is independent of u1} ..., up.
Let the variables ul9 . . . , up be represented, in the ordinary way, each by
the real points of an infinite plane. Put xl = e2niUt , . . . , xp = e2™"? ; then to
the finite part of the wa-plane (a=l, "-,p) corresponds the portion of an
avplane lying between a circle Fa of indefinitely great but finite radius Ra,
whose centre is at xa = 0, and a circle ja of indefinitely small but not zero
radius ra, whose centre is at xa = 0. The annulus between these circles may
be denoted by Ta. Let aa be a value for xa represented by a point in the
annulus Ta ; describe a circle (Aa) with centre at aa, which does not cut the
circle ^a ; then for values of aca represented by points in the annulus Ta which
are within the circle (^o), u* may be represented by a series of integral
positive powers of xa — aa ; and by the ordinary method of continuation, the
values of ua for all points within the annulus Ta may be successively re
presented by such series; the most general value of wa, for any value of xa, is
of the form xa + m, where m is an integer. Thus, in virtue of the definition,
(f)(u1, ...,up) is a single-valued, and analytical, function of the variables
xl} ...,xp, which is finite and continuous for values represented by points
within the annuli Tl,...,Tp and upon the boundaries of these. So considered,
denote it by -\Jr (^ , . . . , xp).
Take now the integral
1
wherein xl,...,xp are definite values such as are represented by points
respectively within the annuli T1} ..., Tp; let its value be formed in two
ways;
(i) let the variable ta be taken counter-clockwise round the circum
ference Fa and clockwise round the circumference 7a(a = l, ...,j9); when ta is
upon the circumference F0 put
JL -1. Xd *™a. x* ft
— "1 ^ "•" ~~Q i *• ^ -j-i >
ta — Xa Va. ta ta »-~ "a
when ta is upon the circumference 7a put
4+..-) = -
. T 2^ l~ "I . ' .*tt+l '
ta.-X,. \Xa Xa X. ' fca=~C°^ft
then the integral is equal to
where dZa represents an element dta taken counter-clockwise along the
circumference F0, and dza represents an element dta taken clockwise along
281] EXPONENTIAL SERIES. 445
the circumference ya ; since the component series are uniformly and absolutely
convergent, this is the same as
Mi ftp
i oo oo r r r T*.. T
1 2 ... 2 //..,/*(« .W^Sr^*...^
^.iiTn^r «1=-oo np=-<x>JJ J ti ...tp
where for 4 the course of integration is a single complete circuit coincident
with F0 when na is positive or zero, and a single complete circuit coincident
with 70 when na is negative, the directions in both cases being counter
clockwise ; thus we obtain, as the value of the integral,
00 00
2-^ A "I nP
2* A. n,, ... , np &\ . . . Xp ,
n\= — °o MJ>= — oo
where
and the course of integration for ta may be taken to be any circumference
concentric with Fa and ya, not lying outside the region enclosed by them ;
(ii) let the variable ta be taken round a small circle, of radius pa,
whose centre is at the point representing xa (a = 1, . . . , w) ; putting
we obtain, as the value of the integral, -v/r (as1} ..., xp).
The values of the integral obtained in these two ways are equal*; thus
we have
00 00
<p (Ui, . . . , Up) = _* ... — < •"-«!, ..., nlt & »
w,= — oo rip=— cc
where
/•I /"I
^!MI n = ... g-2«(»»1«1+...+«pMp) ^(-Wi) ; i^)dtt, ... dup.
Jo Jo
By the nature of the proof this series is absolutely, and for all finite
values of ult . . . , up , uniformly convergent. If ua = va + iw* (a. = 1, . . . , p), and
M be an upper limit to the value of the modulus of (f> (MJ , . . . , up) for assigned
finite upper limits of wl} ..., wp, given suppose by | wa \ ^> Wa, we have
where Na =
Ex. i. Prove that
— r
OW i n
as "
Ex. ii. In the notation of § 174, Chap. X.,
* Cf., for instance, Forsyth, Theory of Functions, p. 47. The reader may also find it of
interest to compare Kronecker, Vorlesungen iiber Integrate (Leipzig, 1894), p. 177, and
Pringsheim, Math. Annal. XLVII. (1896), p. 121, ff.
446 LINEAR INDEPENDENCE OF THE 2^ THETA FUNCTIONS [282
282. Further it is useful to remark that the series obtained in § 281 is
necessarily unique ; in other words there can exist no relation of the form
00 00
V V A ni Mp f\
2, ... 2, AHi> ^ttlpXi ...Xp = 0,
«!=— 00 Wp=-00
valid for all values of xlt ..., xp which are given, in the notation of § 281, by
^a<|^0|< -Ra, unless each of Ani< ___>np be zero. For multiplying this equation
by #1 ' ... xp dx1 ... dxp, and integrating in regard to xa round a circle,
centre at xa = 0, of radius lying between ra and Ra, (a= 1, ..., p), we obtain
An important corollary can be deduced. We have remarked (§ 175,
Chap. X.) on the existence of 2^ theta functions with half-integer character
istics ; it is obvious now that these functions are not connected by any linear
equation in which the coefficients are independent of the arguments. For an
equation
22P oo oo
2 Cgg kg 2 ... S e^«(n+i*«>+&(»-H*«)*+ttrgr«(n+i*,) = 0,
S = l »»i = — °° Wp=-oo
where the notation is as in § 174, Chap. X., and ks, gs denote rows of p
quantities each either 0 or 1, can be put into the form
i .....
NI = - oo Np=-<x>
where 27n'£71, ..., ZiriUp are the quantities denoted by hu, A^ ..... Np is
given by
ANl, .... Np = 2 Cffii
!)>
where the summation includes 2^ terms, and N1} ..., Np take the values
arising, by the various values of n and ks, for the quantities 2n + ks ; it is clear
that the aggregate of the values taken by 2n + ks when n denotes a row of p
unrestricted integers, and ks a row of quantities each restricted to be either 0
or 1, is that of a row of unrestricted integers.
Hence by the result obtained above it follows that -Ajr,, .... jyp = 0, for all
values of n and ks. Therefore, if A, denote a row of arbitrarily chosen
quantities, each either 0 or 1, we have
= 0 ;
adding the 2^ equations of this form in which the elements of n are each
either 0 or 1, the value of ks being the same for all, we have
where n,l, ..., /JLP are the elements of the row letter /JL given by fj. = gs + \;
the product (1 + ein^} ... (1 +eiwfj-") is zero unless all of /^, ..., yuy are even,
284] WITH HALF-INTEGER CHARACTERISTICS. 447
that is, unless every element of gg is equal to the corresponding element of X.
Hence we infer that C^ k, = 0 ; and therefore, as \ is arbitrary, that all the 22^
coefficients C(Ji t ^ are zero.
Similarly the r2? possible theta functions whose characteristics are rth
parts of unity are linearly independent.
283. Another* proof that the 2'* theta functions with half-integer
characteristics are linearly independent may conveniently be given here : we
have (§ 190), if m and q be integral,
and therefore if A; be integral and Q' = q' + k', Q = q + k,
Therefore a relation
8=1
leads to
2 CU™ <™<2*'-™'<W * (u • fa) = 0,
8=1
where Qg = qs + k, Qg' = qs' + k' ; in this equation let (m, m') take in turn all
the 2^ possible values in which each element of m and m' is either 0 or 1 ;
then as
•^gwUmQg'-m'Q,) t _ []_ + gfilQ,'),-] ... [1 + e«(Q«')p] [1 -f g-«(Q«)i] ... [1 + e
is zero unless every one of the elements (Q/X, ..., (Q«)^ is an even integer,
that is, unless qs = k, qg'= k', we have
2 £.«•<••*'-*•«« ^ (M ; fa) = 2*pCk<$ (u • P) = 0 ;
TO 8 = 1
thus, for any arbitrary characteristic (k, k'), C^ = 0. Thus all the coefficients
in the assumed relation are zero.
284. We suppose now that we have four matrices o>, &>', 77, 77', each of p
rows and columns, which satisfy the conditions, (i) that the determinant of o>
is not zero, (ii) that the matrix a)"1^' is symmetrical, (iii) that, for real values
of !&!,..., np, the quadratic form co^ca'n2 has its imaginary part positive^,
(iv) that the matrix 7]w~l is symmetrical, (v) that 77' = v)w~l<o — ^ 7ria)~l ; then
the relations (B) of § 140, Chap. VII., are satisfied ; we put a = ^rjco"1,
h = %iri(D~~l, b = Triarla>, so that (cf. Chap. X., § 190)
77 = 2aeo, t]' = 2aa>' — h, hat = \tn, hco' = ^b ;
* Frobenius, Crelle, LXXXIX. (1880), p. 200.
t Which requires that the imaginary part of the matrix <a~lu' has not a vanishing de
terminant.
448 EXPRESSION FOR A GENERAL [284
as in § 190 we use the abbreviation
\m (u) = Hm (u + £Hm)
where
Hm = 2r)m + 2r)'m', Hm = 2&>m + 2&>'?n/.
We have shewn (§ 190) that a theta function ^ (u, q) satisfies the
equation
^ (u + Slm,q) = e*mW+*ri(™/-»W ^ (u> g,),
m and m each denoting a row of integers; it follows therefore that, when
m, m each denotes a row of integers, the product of r theta functions,
n (u) =* (u, q(») * (u, ?(2>) ...... *t(u, q^l
satisfies the equation
n (U + nm) = e»*m(W)+2«<mQ'-m'Q) n (u),
wherein Qit Q/ are, for i=l, 2, ..., p, the sums of the corresponding com
ponents of the characteristics denoted by q{l), ..., q(r).
Conversely*, Q, Q' denoting any assigned rows of p real rational
quantities, we proceed to obtain the most general form of single- valued,
integral, and analytical function, II (u), which, for all integral values of
m and m, satisfies the equation just set down. We suppose r to be an
integer, which we afterwards take positive. Under the assigned conditions
for the matrices o>, &>', 77, ?/, such a function will be called a theta function
of order r, with the associated constants 2«, 2&>', 2?;, 2V, and the characteristic
(Q, Q'l
Denoting the function 3(u; Q), of § 189, either by 3 (it; 2o>, 2«', fy, 2i/; <?, (?) or
3(u; a,b,h; Q, Q'), the function 3(u; 2a>/r, 2w', 2rj, 2n/ ; Q, Q'/r) is a theta function of
the first order with the associated constants 2o>/r, 2o>', 2?;, 2rr)', and (Q, Q'/r) for charac
teristic ; increasing u by 2<am + 2a>'m', where m, m' are integral, the function is multiplied
by a factor which characterises it also as a theta function of order r, with the associated
constants 2«, 2<o', 2rj, ty and ($, <7) for characteristic. We have, also,
B(u; ra, rb, rh) = $(u; — , 2«', 2^, irVj-^fnt; 2a, 2>-o>'; -^, 2^'J = sfra; ", A, rfcj,
where the omitted characteristic is the same for each.
Let ki be the least positive integer such that kiQJ is an integer, =/j, say ;
denote the matrix of p rows and columns, of which every element is zero
except those in the diagonal, which, in order, are kl} kz> .... kp, by k; the
inverse matrix Ar1 is obtained from this by replacing k^ ... respectively by
* Hermite, Compt. Rend. t. XL. (1855), and a letter from Brioschi to Hermite, ibid. t. XLVII.
Schottky, Abriss einer Theorie der AbeVschen Functionen von drei Variabeln (Leipzig, 1880), p. 5.
The investigation of § 284 is analogous to that of Clebsch and Gordan, Abel. Funct., pp. 190, ff.
The investigation of § 285 is analogous to that given by Schottky. Of. KSnigsberger, Crelle, LXIV.
(1865), p. 28.
284] THETA FUNCTION OF ORDER r. 449
l/&i , ... ; iu place of the arguments u introduce arguments v determined by
the p equations
hiilu1 + +hi>pup = kivi, (i = 1 , . . . , p),
which we write hu = kv; then, by the equations hw = ^TTI, ha)' = £6, it follows
that the increments of the arguments v when the arguments u are increased
by the quantities constituting the p rows of a period ftw, are given by the p
rows of Um defined by
kUm = Trim + bmf ;
we shall denote the right-hand side of this equation by Tm; thus
Now we have
a (u + nm)2 — auz =
and, since* the matrix a is symmetrical, and Hm = 2aflm - 2hm', this is
equal to
2atlmu + a£lm = 2aHm (u + £flm) = (Hm + 2hm') (u
and therefore equal to
\n (u) + Trimm' + 2hum' + h£lmm'
or
^•m (u) + Trimm' + 2kvm' + Tmm' ;
thus, by the definition equation for the function II (u), we have
e n (u + n,n) = Q—rau n (ii) .
therefore, if Q (v) denote e~rau* U (u),
G (11 -4- JT \ — f) I w\ a—r[Trimm'+
W \v -\- um) — \j \u) e L
now let m = 0, and m = ks, where s denotes a row of integers s1} . . . , sp ; then
mQ' = ksQ =&1s1Q/+ + kpspQp =skQ', = sf, is also a row of integers;
and Um = Trik~lm + k~*bm' = TTIS ; thus we have
Q (v + Tris) = Q (v),
or, what is the same thing, the function Q(v) is periodic for each of the
arguments vlt ...,««, separately, the period being TTI; it follows then (§ 281)
that the function is expressible as an infinite series of terms of the form
tfn1,n2,...,«pe2(n'l'1+"'~Hv'p)> where nlt ..., np are summation letters, each of
which, independently of the others, takes all integral values from - x to
+ 00, and the coefficients (?„, np are independent of vlt ..., vn. This we
denote by putting
Q(v) = e-rau*n(u) = 2Cne™\
To this relation, for the purpose of obtaining the values of the coefficients
* By a fundamental matrix equation, if M be any matrix of p rows and columns, and «, 7- be
row letters of p elements, n u v = fi. v u.
B- 29
450 EXPRESSION OF A GENERAL [284
Cn, we apply the equation, obtained above, which expresses the ratio to
Q (v) of Q (v + Um) or Q (v + k~lTm) ; thence we have
26'
in this equation, corresponding to a term of the left-hand side given by the
summation letter n, consider the term of the right-hand side for which the
summation letter s is such that
Si = n{ + rkinii, (i = 1, 2, . . . , p) ;
thus s = n + rkm', and 2vtSi = 2^w; -I- 2rkiVtmi', or 2vs = 2wn + 2rkvmf ; hence
we obtain
therefore, equating coefficients of products of the same powers of the
quantities e2"1, . . . , e?°p, we have
n _ n 0->k-lYmn+r(irimm'+Ymm')—zni(m<X—m'Q)
^n+rkm' — ^n • & >
and this equation holds for all values of the integers denoted by n, m, m.
By taking the particular case of this equation in which the integers m'
are all zero we infer that the quantity
— . k-irrmn — mQ', = —. k~l (Trim) n — mQ', = S ms ( r ns — Q,']
TTl Til s = l \KS /
must be an integer for all integral values of the numbers rag and ns ; therefore
the only values of the integers n which occur are those for which the
numbers (ng -k8Qs')/ks are integers; thus, by the definition of ks, we may put
n =f+ kN, N denoting a row of integers, and/= kQ'.
With this value we have
k~lftmn — k~l (Trim) n = k~l (bmf) n = k~ln (bin'} = k~ln . bin'
= (&-'/+ N) . bm = (Q' + N) . bm' = bm (Q' + N) ;
hence, as mQ' = k~lmn — mN, the equation connecting Cn and Cn+rkm> becomes
— g
e27rtVmm' being equal to unity because r is an integer, and bm' Q' — bQ'm = bQ'm ;
therefore
--Hm'r-fAT)2 „ ~1&^2 ov ,
e * Cf+k(m.r+N} = e * Gf+kN . e^'n ,
TQ being mQ + bQ', or
284] THETA FUNCTION OF ORDER r. 451
thus, if the right-hand side of this equation be denoted by DN, we have, for
every integral value of m, DN+rm> = DN ; therefore every quantity D is equal
to a quantity D for which the suffix is a row of positive integers (which may
be zero) each less than the numerical value of the integer r. If then p be
the numerical value of r, the series breaks up into a sum of pP series; let J9M
be the coefficient, in one of these series, in which the integers p are less than
p ; then the values of the integers N occurring in this series are given by
JV" = fj, + rM, M being a row of integers, which, as appears from the work,
may be any between — oo and oo ; and the general term of Q (v) is
for k.(Q' + N)v = kv(Q' + N)=kv(Q' + N) = hu (Q' + N) ; thus the general
term is
now, as Tg = TriQ + bQ', and b is a symmetrical matrix, the quantity
is immediately seen to be equal to
rb M + thi?1 + 2«<3
r r
therefore the general term of n (u), or erau~ Q (v), with the coefficient DM, is
e++x, where
= rait + Zrhu (j\f + ?-±-£} + rb(M+ ±^Y + 2-rriQ ( M +
\ r I \ r J V T
' V M M
and this is the general term of the function
Q' + n
where ^ denotes a theta function differing only from that before represented
(§ 189, Chap. X.) by ^, in the change of the matrices a, b, h respectively into
ra, rb, rh ; the condition for the convergence of the series ^ requires that r
be positive ; thus p = r ; recalling the formulae
we see, as already remarked on p. 448, that, instead of
w, &>', 77, ij',
29—2
452 EXPRESSION OF A GENERAL [284
the quantities to be associated with the function ^ are
6)
- , a , 97, rrj ,
with this notation then we may write, as the necessary form of the function
it(«X
n (u) = 2tf „* (u • Q, Q±P]
~ \ r )
wherein K^, = D^e r r is an unspecified constant coefficient, ft denotes
a row of p integers each less than the positive integer r, and the summation
extends to the rp terms that arise by giving to yu, all its possible values.
From this investigation an important corollary can be drawn ; if a single-
valued integral analytical function satisfying the definition equation of the
function II (M) (p. 448), in which r is a positive integer and the quantities
Q, Q' are rational real quantities, be called a theta function of the rth order
with characteristic (Q, Q'), then* any rp + l theta functions of the rth order,
having the same associated quantities 2o>, 2&>', 2rj, 2rj' and the same charac
teristic, or characteristics differing from one another by integers, are connected
by a linear equation or by more than one linear equation, wherein the
coefficients are independent of the arguments u^, ..., up ; and therefore any
of the functions can be expressed linearly by means of the other rp functions,
provided these latter are not themselves linearly connected.
For the determining equation satisfied by II (u) is still satisfied if, in
place of the characteristic (Q, Q'), we put (Q + N, Q' + N'), N and N' each
denoting a row of p integers; and if
p + N' = v (mod. r), say p + N' = v + rL',
we have (§ 190, Chap. X.)
and therefore
where Hv = K^nlN~^ ; and the aggregate of the rP values of — - is the
CV -i.
same as that of the values of — .
r
Thus any rp + 1 theta functions of the rth order, with the same charac
teristic, or characteristics differing only by integers, and associated with the
* The theorem is attributed to Hermite : cf. Compt. Rendus, t. XL. (1855), p. 428.
285] THETA FUNCTION OF ORDER r. 453
same quantities 2<w, 2o>', 2??, 2tj', are all expressible as linear functions of
the same r# quantities ^ ( u ; Q, — — - ) with coefficients independent of
\ r I
u1} ..., up. Hence the theorem follows as enunciated.
Ex. i. Prove that the r" functions 3 (u ; Q, ^-^-\ are linearly independent (§ 282).
Ex. ii. The function $ (u + a ; Q) $ (u-a • Q) is a theta function of order 2 with
(2Q, 2<7) as characteristic. Hence, if 2P+1 values for the argument a be taken, the
resulting functions are connected by a linear relation.
For example, when p = l, we have the equation
o-2 (a) a- (u -b)<r(u + b)- a* (b) or (u - a) <r (u + a) = a3 (u) . <r (a-b)tr (a + b).
Ex. iii. The function 9 (ru, Q) is a theta function of order r2 with (rQ, r(?) as
characteristic. Prove that if £ denote a theta function with the associated constants
w> r2&)'> ^ ' •/» m place of a), w', T), T)' respectively, then we have the equations
where the summation letters /*, j/ are row letters of p elements all less than r, and each
summation contains f? terms.
Ex. iv. The product of k theta functions, with different characteristics,
is a theta function of order k for which the quantities
2
Lr=
2 $r)- 2r/ 2 wM, 2 ^'(r> + 2^ 2
r=l r=l
enter as characteristic. Thus a simple case is when uW + . .. + uW = 0.
For p = l a linear equation connects the five functions
4
4 4
n a(u+ut), n ir (v+«|+4»X u r(«+«t+»7, n
!=1 =
Ex. v. Any (p + 2) theta functions of order r, for which the characteristic and the
associated constants w, o>', ,, ^' are the same, are connected by an equation of the form
P=0, where P is an integral homogeneous polynomial in the theta functions. For the
number of terms in such a polynomial, of degree N, is greater than (Nrf, when N is taken
great enough. That such an equation does not generally hold for (p + l) theta functions
may be proved by the consideration of particular cases.
285. The following, though partly based on the investigation already
given, affords an instructive view of the theorem of § 284.
Slightly modifying a notation previously used, we define a quantity,
depending on the fundamental matrices <a, &>', rj, r}', by the equation
X (u ; P, P) = HP (U + $np) - -n-iPP'
= (2r)P + 27/P') (U + 0)P + 0)'P') - TTiPP,
454 ANOTHER PROOF [285
where P, P' each denotes a row of p arbitrary quantities. The corresponding
quantity arising when, in place of &>, &/, 77, if we take other matrices <o(1),
w'w, 77(1), ?/(1) may be denoted by X(1) (u; P, P'). With this notation, and in
case
are respectively
,
- , eo , 77 , rrj ,
r
where r is an arbitrary positive integer, we have the following identity
r 2<» *r /i
r\\u-\ -- s ; Jv , m
= X<» [M + n^ ; A;, 0] + X'1* [u ; m, m'] - X*1' [M ; s, 0] -
where s, N, m, m, k each denotes a row of p arbitrary quantities subject to
the relation
s + rN = m+k;
this the reader can easily verify ; it is a corollary from the result of Ex. ii.,
§190.
Let the abbreviation R (u ; /) be denned by the equation
R (u ; /) = SeT2* («+ v) -2"ifr n (u + 2« -V
fc V T>
^ -\(1)[tt;/(-, 0]-2iri/- ,-r / . 0 M
= Ze f II M + 2&) - ,
« V r;
wherein A; denotes a row of p positive integers each less than r, and the
summation extends to all the rP values of k thus arising, / is a row of p
arbitrary quantities, and II (u) denotes any theta function of order r.
Consider now the value of R (u + fl'i' ; /) ; by definition we have
n rt* + 2« + n) =n
therefore, if m + k = s (mod. r), say m + k = s + rN, we have, by the defin
ition equation (§ 284) satisfied by II (M),
where (Q, Q') is the characteristic of II (u), and hence
R(u+ n|l'; /) = 2e*II (M+ 2ft><1>s),
in which
.„*
r
285] OF THE EXPRESSION. 455
by the identity quoted at the beginning of this Article, i/r can also be put
into the form
i/r = \a> [u ; ra, m'] - X"> [u ; s, 0] - Zirim'k - Zirif- + Z-rri (NQ1 - m'Q),
= X(1) [u ; Tii, m'] - V" [it ; s, 0] - 2-jrim'k - 2-irim'Q + 2-iriN(Q' -/)
" s
in the definition equation for II (u), the letters m, m denote integers ; and
k has been taken to denote integers ; if further f be chosen so that Q' —f is
a row of integers, we have, since, by definition, N denotes a row of integers,
_(D A(1)[M; m, m'] + 2iri(mf-m'Q) -\(1) (u ; s, 0) -Zirifi ,_,
12 (M + ftjf ; /) = e \ r ' 2e r II (u + 2<w(1> s)
= e
; m, m']+2iri ^w ~-m'Q\
> R (u • f).
Hence R(u; f) satisfies a determining equation of precisely the same
form as that satisfied by II (u), the only change being in the substitution of
— , w', 77, rrj' respectively for <w, <y', 77, 77' ; so* considered R(u; f) is a theta
function of the first order with (Q, - J as characteristic; putting, in ac
cordance with the definition of f above, f= Q' + fj,, where p is a row of p
integers, we therefore have, by § 284,
R (u ;?+»)- KQ^ * (u ; Q, Q^} , = Kq+. * (ru • * , h, rb ; *?} ,
\ r / \ r Q /
(p. 448) where Kq+lli is a quantity independent of u, and ^ is the same theta
function as that previously so denoted (§ 284), having, in place of the usual
matrices a, 6, h, respectively ra, rb, rh.
Remarking now that the series
wherein p denotes a row of p integers (including zero), each less than r, and
the summation extends to all the rp terms thus arising, is equal to r? when
the p integers denoted by k are all zero, and is otherwise zero, we infer that
the sum
1 .
which, by the definition of R (u, /), putting /= Q' + p, is equal to
i 2 [.-*"*" *••'-««'* n («+ a.^
B (« ; /) may also be regarded as a theta function of order r, with the associated constants
2tu, 2u', 2?;, 217' and characteristic (Q, f).
456 EXAMPLES OF THE APPLICATION [285
is, in fact, equal to II (u). Hence as before we have the equation
286. Ex. i. Suppose that m is an even half-integer characteristic, and that
are s, = W, half-integer characteristics such that the characteristic formed by adding the
three characteristics m, ait a,- is always odd, when i is not equal to j. Thus when m
is an integral, or zero, characteristic, the condition is that the characteristic formed by
adding two different characteristics at, 0,-may be odd. The characteristic whose elements
are formed by the addition of the elements of two characteristics a, b may be denoted by
a + b ; when the elements of a + b are reduced, by the subtraction of integers, to being less
than unity and positive (or zero), the reduced characteristic may be denoted by ab.
For instance when p = 2, if a, ft, y denote any three odd characteristics, so that * the
characteristic afty is even, and if /* be any characteristic whatever, characteristics satis
fying the required conditions are given by taking m, alt a.2, a3, a4 respectively equal to
afty, p., fifty, pya, ^aft ; in either case a characteristic mo^o,- is one of the three a, ft, y and is
therefore odd.
When jo = 3, corresponding to any even characteristic m, we can in 8 ways take seven
other characteristics a, ft, y, K, X, p, v, such that the combinations a, ft, y, K, X, p, v, ma/3,
man, mX/i constitute all the 28 existent odd characteristics ; this is proved in chapter
XVII. ; examples have already been given, on page 309. Hence characteristics satisfying
the conditions here required are given by taking
m, «19 «2> «3> -•> as
respectively equal to
m, m, a, ft, ..., v.
Now, by § 284, every 2" + 1 theta functions of the second order, with the same periods
and the same characteristic, are connected by a linear equation. Hence, if p, q, r denote
arbitrary half-integer characteristics, and v, w be arbitrary arguments, there exists an
equation of the form
A9(u + w; q)S(u-iv; r)= 2
A~*=l
wherein A, A^ are independent of u ; for each of the functions involved is of the second
order, as a function of u, and of characteristic q + r.
We determine the coefficients A^ by adding a half period to the argument u ; for u
put u + Qm-aj-p ; then by the formula
S (u + QP, <?) = / <"; P] -*"iP'q B (u ; P+q),
where
X (u ; 1>] = HP (u +£aP) - viPl",
noticing, what is easy to verify, that
(u-v; P)-\ (u+w; P}-\(u-iv\ P) = 0
As the reader may verify from the table of § 204 ; a proof occurs in Chap. XVII.
286] OF THE THEOREM. 457
we obtain
= 2 A^[u + v; (m-aj-a^ + q + r-2p)]$[u-v; (m - a,- -f «
But since m-aj + a^ (which, save for integers, is the characteristic maja^ is an odd
characteristic when./ is not the same as X, we can hence infer, putting u = v, that
Hence the form of the relation is entirely determined. The result can be put into
various different shapes according to need. Denoting the characteristic m + q + r
momentarily by k, so that k consists of two rows, each of p half-integers, and similarly
denoting the characteristic (t^+p momentarily by aA, and using the formula for integral M,
el$(u; q),
we have
e-'a^3(Zv; k);
we shall denote the right-hand side of this equation by
e-4rf<aA+P)(,»'+9'+r') .jra,, ; (m+q + r)] ;
hence the final equation can be put into the form
; m]
It may be remarked that, with the notation of Chap. XI., if 615 ..., bp be any finite
branch places, and Ar denote the characteristic associated with the half-period ub>--a, and
we take for the characteristics a:, ..., «g the 2» characteristics A, AA^ ... Ak, formed by
adding an arbitrary half-integer characteristic A to the combinations of not more than p
of the characteristics A1, ..., Ap, and take for the characteristic m the characteristic
associated with the half-period iibi>ai + ... + ubp>ap, then each of the hyperelliptic functions
5(0; raaiO,-) vanishes (§ 206), though the characteristic rn.^^ is not necessarily odd.
Hence the formula here obtained holds for any hyperelliptic case when m, alt ..., a,, have
the specified values.
Ex. ii. When p = 2, denoting three odd characteristics by a, /3, y, we can in Ex. i. take
P, q, r, m, «i, «2> «a> ai
respectively equal to
a^y, q, 0, a/3y, 0, /3y, ya, a/3,
wherein 0 denotes the characteristic of which all the elements are zero, and £y denotes
the reduced characteristic obtained* by adding the characteristics # and y. Then the
general formula of Ex. i. becomes, putting v = 0 and retaining the notation m for the
characteristic a/3y,
3(u+w; q)3(u-w; 0)5(0; q + m)3(0; TO)
«A + '")(-' + «-) 3 (u; q-m-a^3(u; m+a^S(w; q-aj9(w, aA).
So that all the elements of £7 are zero or positive and less than unity.
2
A — 1
458 A GENERAL ADDITION THEOREM [286
EM. iii. As one application of the formula of Ex. ii. we put
'10\ , /10\ , /01\ , /Ol
and therefore
'ION , /oo\ , /oo
m =
hence we find, comparing the table of § 204, and using the formula
where M, =(M^/}, consists of integers, f-Cffi), and Mf = MJ^ + MJ^ that*
\M1M2J V/i/2/
(u-<w; 0) = 56(M-w), 5(0; ? + m) = 5J2 (0), 5(0; m) = 301(0;,
3(u;q-m-a3) = 524(w),5(
5 (M ; ? - m - o4) = - 514(«), 5 (M ; m + eg = - 53 (w), 5 (w ; g- - a4) = 5M(w), 5 (w ; a4) = - 524(w),
all the factors of the form eiwi(a\+ mXm' + q"> being equal to 1 ; by substitution of these
results we therefore obtain
502 (u + w) 55 (u-w) 512 (0) 501 (0) = 51250150256 + 5025651850, + 50152453514+535M5045!J4,
where 512 denotes 512 (w), etc., and 302 denotes 502 (w), etc. ; this agrees with the formula
of §§ 219, 220 (Chap. XI.).
Ex. iv. By putting in the formula of Ex. ii. respectively
obtain the result
which is in agreement with the results of §§ 219, 220.
Dividing the result of Ex. iii. by that of Ex. iv. we obtain an addition formula for the
theta quotient 502 (tt)/56 (w), whereby 9oz(u + w)/$5(u+w) is expressed by theta quotients
with the arguments u and w.
Ex. v. The formula of Ex. ii. may be used in different ways to obtain an expression
for the product 3 (u + w; q) 5(w — w; 0). It is sufficient that the characteristics m and
o+m be even and that the three odd characteristics a, /3, y have the sum m. Thus,
starting with a given characteristic y, we express it, save for a characteristic of integers,
as the sum of two even characteristics, m and q+m, which (unless q be zero) is possible
in three wayst, and then express m as the sum of three odd characteristics, a, /3, y,
which is possible in two waysj; then§ we take ^ = 0, a2 = /37> a3 = yat «4=o£. Taking
,
' ave
* In Weierstrass's reduced characteristic symbol the upper row of elements is positive, and
the lower row negative ; cf. §§ 203, 204, and p. 337, foot-note.
t This is obvious from the table of § 204, or by using the two-letter notation ; for instance
the symbol (a^J^a^) + (a2c) = (a1c1) + (a2c1)=(a1c2) + (a^).
t For example, (ac) = (t^a) + (a2a) + (c^) = (a^) + (cjc) + (cc2). See the final equation of § 201.
The six odd characteristics form a set which is a particular case of sets considered in
chapter XVII.
§ Moreover we may increase M and w by the same half-period. But the additions of the half-
periods P, P + fl3 lead to the same result ; and, when q is one of a, /3, 7, the same result is
obtained by the addition of P + ttm and of P + QTO + i2a.
286] FOR THE CASES p = 2, p = 3. 459
10\ /10\ /00\ /01\ /11\ /00\ /IO
putting m=\ fwvi we may
10
Hence obtain the result
502 (u+w) 55 (M- w) 512 (0) 301 (0)-*M5w5tt5,+3w5il51458+ Vu^^+^^S^,,
where, on the right hand, 512 denotes 512(w), etc., and 502 denotes 502(w), etc. Comparing
this result with the result of Ex. iii., namely
5 (u-w) 512 (0) 501 (0) = ^125010255 + 502^5512501 + 50^2453514 + 53^4504524,
we deduce the remarkable identity
54 (u) 513 (M) 523 (w) 503 ( w) + ^ (w) 5M («) 50 (w) 32 (w)
= 502 (w) 55 (u) 31S (w) 501 (w) +33 (M) 514 (w) 5M (w) 524 (w),
wherein w, w tare arbitrary arguments ; this is one of a set of formulae obtained by
Caspary, to which future reference will be made.
Ex. vi. By taking in Ex. v. the characteristics q, m to be respectively
(• *($'
and resolving m into the sum a+/3 + y in the two ways
respectively, obtain the formulae
502 (M + w) $B (u-w) B0 (0) 52 CO)»^^JA+\^U - 54^i85M5M - 3M3M51S$4)
502 (« + w) 5- (M - w) 50 (0) 32 (0) = 505250255 - SM5W54513 - 5145,503523 + ^5^0, 512,
and the identity
•^34 ^1 ^01 ^12 + °i °13 ^24 ^04 = ^5 ^02 ^0 ^ 2 + ^14 ^3 °03 ^23 '
Putting in this equation w = 0, we obtain a formula quoted without proof on page 340.
Ex. vii. Obtain the two formulae for 502 (?< + w) 55 (u — w) which arise, similarly to
those in Exs. v. vi., by taking for m the characteristic | ( J , the characteristic q being
unaltered.
Ex. viii. Obtain the formulae, for p = 2,
523 (u + w) ^ (u - w) 56 (0) 523 (0; = ^^ia^^ + ^ 5,^5! 5^ - 53525352 - 513512513512,
where the notation is as in Ex. v.
For tables of such formulae the reader may consult Konigsberger, Crelle, LXIV. (1865),
p. 28, and ibid., LXV. (1866), p. 340. Extensive tables are given by Rosenhain, M4m. par
divers Savants, (Paris, 1851), t. XL, p. 443 ; Cayley, Phil. Trans. (London, 1881),
Vol. 171, pp. 948, 964 ; Forsyth, Phil. Tram. (London, 1883), Vol. 173, p. 834.
460 A MORE GENERAL FORMULATION. [286
Ex. ix. We proceed now to apply the formula of Ex. i. to the case p = 3 ; taking the
argument v = 0, the characteristics p, r both zero, and the characteristics TO, CTJ, a2, ...... , as
to be respectively TO, m, a, /3, ...... , f, where a, /3, y, K, X, p, v are seven characteristics
such that the combinations a, /3, y, K, X, p., v, ma/3, TOOK, mX/x are all odd characteristics,
TO being an even characteristic, and removing the negative signs in the characteristics by
such steps'* as
= e- 47rim(a' A + p'+ m') $ (
= e-47rim(p' + a'A)^(M;.
the formula becomes t
8
= 2 e~'Mm«'A. + «'aA>.»(
\=i
In order that the left-hand side of this equation may not vanish, the characteristic
q + m must be even; now it can be shewn that every characteristic (q), except the zero
characteristic, can be resolved into the sum of two even characteristics (TO and q + m)
in ten ways, and that, to every even characteristic (TO) there are 8 ways of forming such
a set as a, /3, y, K, X, /x, v (cf. p. 309, Chap. XI.). Hence, for any characteristic q there
are various ways of forming such an expression of 3(u + w; q)3(u — w; 0) in terms
of theta functions of u and w ; moreover by the addition of the same half-period to u
and w, the form of the right-hand side is altered, while the left-hand side remains
effectively unaltered. In all cases in which q is even we may obtain a formula by
taking TO = 0.
Ex. x. Taking, in Ex. ix., the characteristics q, m both zero, prove in the notation
of § 205, when a, /3, ...... , v are the characteristics there associated with the suffixes
1, 2, ...... , 7, that
3(u+u;)3(u-w)9* = 2 $?(u}S?(w}.
i=0
Prove also, taking m=0, q = ^ ( AAn ) , that 3456 (u + w) 3 (u - w) 5456 9 is equal to
\UUv/
3 (u) 3 (w) Sm (u) 3456 (w) +$t (u) 54 (w) $56 (u) 556 (w) + S5 (u) 9, (w) \4 (u) 364 (w)
+V«) ?•(*
- 37 (u) 57 (w) 3123 (u) $m (w) - ^ (u) 3, (w) 9237 (u} 3237 (w) - 32 (u) $2 (w) 3317 (u) 9317 (w)
-$3(u)93(w)9ia(u)9l27(w),
where 3, #456 denote respectively 9 (0), £456 (0).
Hence we immediately obtain an expression for •9456(«+ w)j§(u + w} in terms of theta
quotients $t (u)/3 (u), 3* (w)/& (w).
Ex. xi. The formula of Ex. i. can by change of notation be put into a more symmetrical
form which has theoretical significance. As before let TO be any half-integer even
characteristic, and let «:, ...... , as be s, =2", half-integer characteristics such that every
* Wherein the notation is that the characteristic p is written (Pl P* Ps } and p' denotes the
\PiPzPs I
row ( PI, p2', p3') ; and similarly for the characteristics m, a^.
t This formula is given by Weber, Theorie der AbeVschen Functionen vom Geschlecht 3
(Berlin, 1876), p. 38.
287] ODD AND EVEN FUNCTIONS. 461
combination mc^o,-, in which i is not equal to ,;', is an odd characteristic ; let /, g, h be
arbitrary half-integer characteristics ; let J denote the matrix of substitution given by
J=\(-\ 1 1 1),
1-111
1 1-1 1
1 1 1-1
and from the arbitrary arguments u, v, w determine other arguments U, V, W, T by the
reciprocal linear equations
(Uit Vit W<, Ti} = J(ui, vit wt, 0), (i = l, 2, , p\
or, as we may write them,
( U, V, \V, T) = J(u, v, w, 0) ;
further determine the new characteristics F, O, ff, K by means of equations of the
form
(F, G,H, K)=J(f,g,h,m),
noticing that there are 2p such sets of four equations, one for every set of corresponding
elements of the characteristics ;
then deduce from the equation of Ex. i. that
3(0; m)3(u;f)3(t>; g)3(w; h)
ZP
A = l A ' A '
Putting m = 0, we derive the formula
3(0; Q)3(v + w; g+h)3(w + u; h+f) 3(u + v; f+g)
= 2 3(u+v + w; f+g + h + a,,) 3(u; /— a.) 3(v: a — a ) 3 (w h — a )
A=l
wherein u, v, w are any arguments and/, g, h are any half-integer characteristics.
Ex. xii. Deduce from Ex. i. that when jo = 2 there are twenty sets of four theta
functions, three of them odd and one even, such that the square of any theta function can
be expressed linearly by the squares of these four.
287. The number, r?, of terms in the expansion of II (u) may be
expected to reduce in particular cases by the vanishing of some coefficients
on the right-hand side. We proceed to shew* that this is the case, for
instance, when II (u) is either an odd function, or an even function of the
arguments u. We prove first that a necessary condition for this is that the
characteristic (Q, Q') consist of half-integers.
For, if II (- u) = ell (u), where e is + 1 or - 1, the equation
n<ti
gives
Schottky, Abrins einer Theorie der Abel'nchen Functional von drn Varialeln (Leipzfg, 1880).
402 NUMBER OF LINEARLY INDEPENDENT FUNCTIONS [287
while, the left-hand side of this equation is, by the same fundamental
equation, equal to
€er\-m{-u)-2m(mQ'-m'Q) H (— w) '
hence, for all values of the integers m, m', the expression
r |>m (M) - \_m (- M)] + 4ffiri (mQf - m'Q)
must be an integral multiple of 2?™ ; since, however,
Xm (u) = Hm (a + £Hm) - irimm = X_m (- u),
this requires that 2 (raQ' - m'Q) be an integer ; thus 2Q, 2Q' are necessarily
integers.
Suppose now that Q, Q' are half-integers; denote them by q, q ; and
suppose that II (u) = ell (- u), where e is +1 or - 1. Then from the
equation
T
since, for any characteristic, S- (u, q) = ^ (— zt, — q), we obtain
(M) = eH (- M) = eSJf^ (- w ; q, ^-] = eZK,,.* (u ; - q, -
/A \ ^'/u\
* L 2
^ t*; q —
where v is a row of positive integers, each less than r, so chosen that
'), (mod. r) ;
thus the aggregate of the values of v is the same as the aggregate of the
values of p,\ therefore, by the formula (§ 190), ^(u-, q + M, q' + M' )
. ^ q'^ wnerein M, M' are integers, we have
„ . . ^T. -- V
= H (u) = eS^e ' ^ f « ; q,
comparing these two forms for II (u) we see that in the formula
the values of /A that arise may be divided into two sets ; (i) those for which
2/4 + 2q' = 0 (mod. r) ; for such terms the value of v defined by the previously
written congruence is equal to /*, and the transformation effected with the
help of the congruence only reproduces the term to which it is applied ; thus,
_ fX + g'
for all such values of /j, which occur, e r is equal to e ; (ii) those terms
288] ALL ODD OR ALL EVEN-
_ v+V
for which 2/z, + 2g' $ ° (mod- r) 5 for such terms ^ = e^e r ' Hence
on the whole II (u) can be put into the form
where the first summation extends to those values of p for which
2/A + 2q' = 0 (mod. r), and the second summation extends to half those values
of fj, for which 2//, + 2q' $ 0 (mod. r). The single term
which can also be written in the form
is even or odd according as II (u) is even or odd ; and this is also true for the
term ^ (u ; q, — — J arising when 2/u + 2g' = 0 (mod. r).
Hence if x be the number of values of p,, incongruent for modulus r,
which satisfy the congruence 2/* + 2^' = 0 (mod. r\ and y be the number of
-*ir' ^
these solutions for which also the condition e r = e is satisfied, the
number of undetermined coefficients in II (u) is reduced to, at most,
288. We proceed now to find x and y ; we notice that y vanishes when
x vanishes, for the terms whose number is y are chosen from among possible
terms whose number is x. The result is that when r is even and the
characteristic (q, q) is integer or zero, and II (— 11) = ell (u), the number of
terms in II (u) is ^rf + 2p~1e; while, when r is odd, or when r is even and
the half-integer characteristic (q, q') does not consist wholly of integers, or
zeros, the number of terms in II (u) is ^ r? + £ [1 — (— )r] ee4™59'.
Suppose r is even ; then the congruence 2/i + 2q = 0 (mod. r) is satisfied
by taking p. = M - — q, and in no other way, M denoting a row of p arbitrary
z
integers. Thus unless q' consists of integers, x is zero, and therefore, as
remarked above, y is zero, and the number of terms in II (11) is \rp. While,
when q' is integral, the incongruent values for /i (modulus r) are obtained by
taking the incongruent values for M for modulus 2, in number 2^ ; in that
_4-io^t?'
case x = 2p ; the condition e r = e is the same as e~ZwiqM ' = e ; when q is
integral, this is satisfied by all the 2P values of Jl/, or by no values of M,
according as e is -f 1 or is — 1 ; in both cases y = '2^~l (1 + e) ; when q is not
464 NUMBER OF ODD OR EVEN FUNCTIONS. [288
integral, p — 1 of the elements of M can be taken arbitrarily and the con
dition e-2«?.af=e determines the other element, so that y = 2?"1. Thus,
when r is even, we have
(1) when q, q' are both rows of integers (including zero), ac = %p,
y = 2*'~1 (1 + e), and the number of terms in II (u) is
2"-1 (1 + e) + £ (rf - 2») = £ 7* + %>~l e,
as stated, there being \ rp + 2^-1 terms when II (u) is an even function, and
^rp — %>-i terms when II (u) is an odd function ;
(2) when q' is integral, and q is not integral, # = 2p, y = 2*"1, and there
fore the number of terms in II (u) is
2?-i + %(rP - 2*) = rP,
in accordance with the result stated ;
(3) when q' is not integral, both x and y are zero, and the number of
terms is ^r?, also agreeing with the given formula.
Suppose now that r is odd, then the equation
rM-2q' M-2q'
2/1, + 2q = rM, or p = ^-±- , = integer + 1-4 ,
wherein Jlf is a row of integers, requires M to have the form 2q' + 22V, where
.2V is a row of integers, and therefore
this equation, since p consists of positive integers all less than r, determines
the value of N uniquely ; hence x = 1. The condition
determines y = 1 or y = 0 according as ee^iqq' = + 1 or = — 1 ; hence the
number of terms in II (u) is
-, or
according as ee^iqq' = + 1 or — 1 ; this agrees with the given result when r is
odd, the number of terms being always one of the numbers £(r*> + 1).
289. It follows from the investigation just given that if we take pro
ducts of theta functions, forming odd or even theta functions of order r, with
the same half-integer characteristic (q, q), and associated with the same
constants 2o>, 2&/. 2?/, 2?/, then when r is even, the number of these which
are linearly independent is, at most, | T? + ^>~l e when the characteristic is
integral or zero, and is otherwise \r? ; while, when r is odd, the number
which are linearly independent is, at most, | (rP+ee^w'), e being ± 1 accord
ing as the products are even or odd functions.
289] GOPEL'S BIQUADRATIC RELATION. 465
Ex. i. In case jt? = 2 there are six odd characteristics, and the sum of any three of
them is even*, as the reader can easily verify by the table of page 303. Let a, /3, y, 8, e, f
denote the odd characteristics, in any order, and let a/3y denote the characteristic formed
by adding the characteristics a, ;3, y. Then the product
n(u) = 3 (u, a) 3 (u, ft) 3 (u, y) 3 (u, afty)
is an odd theta function of the fourth order with integral characteristic. Hence this
product can be written in the form
where p. has the 42 values arising by giving to each of the two elements of /*, independently
of the other, the values 0, 1, 2, 3. Changing the sign of u we have
where v is chosen so that
^ + i/ = 0(mod. 4).
This congruence gives 16 values of v corresponding to the 16 values of /x; of these
there are 4 values for which n=v and 2/x=0 (mod. 4) ; these are the values
M = (0,0), (0,2), (2,0), (2,2),
greater values for the elements of /* being excluded by the condition that these elements
must be less than 4. We have by the formula (§ 190) 3 (u; q + M) = eZlriMi' 3 (u),
comparing this with the original formula for n (u}, we see that
so that the terms in the original formula for n(u) for which v=p are absent, and the
remaining twelve terms may be arranged as six terms in the form
where the summation extends to the following values of /*,
M = (0, 1), (1,0), (1,1), (1,2), (1,3), (2,3);
these values may be interchanged respectively with
M = (0,3), (3,0), (3,3), (3,2), (3,1), (2,1),
if a proper corresponding change be made in the coefficients A.^.
The number 6 is that obtained from the formula $i* + 2p-le, by putting r = 4,
f=-l, p = 2.
Ex. ii. In case p = 2, denoting the odd characteristics by a, ft, y, 8, t, £, and the sum
of two of them, say a and /3, by a£, and so on, each of the four products
3 (11, a) 3 (u, oefl, 3 (u, ft) 3 (u, /3,f), 3 (u, y) 3 (u, y({), 3 (u, d) 3 (u,
or, in Weierstrass's notation, if a, ft, y, 8, e, f be taken in the order in which they occur in
the table of page 303, each of the products
* This is a particular case of a result obtained in chapter XVII.
"• 30
466 GOPEL'S BIQUADRATIC RELATION. [289
is an odd theta function of order 2, and of characteristic differing only by integers
from the characteristic denoted by «£, or, in the arrangement here taken, £ ( j ; thus
any three of these products are connected by a linear equation whose coefficients do not
depend upon u.
Similarly each of the products
3 (u, aSe) 3 (u, aSO, -9 (^ 08*) 3 («, /Mflt 5 (u, 78f) 3 (u, ySf), * (u, 0 3 (u, f),
or, in Weierstrass's notation, if a, £, y, 8, e, f be taken in the order in which they occur
in the table of p. 303, each of the products
314(«)V»), 3oi(*)V»), SiM^M, Si3(tt)S8(tt),
is an even theta function of order 2, and of characteristic differing only by integers
from the characteristic denoted by e£, or, in the arrangement here taken, £ ( j ; thus
any three of these products are connected by a linear equation whose coefficients do not
depend upon u.
Ex. iii. For p = 2 the number of linearly independent even theta functions of the
fourth order and of integral characteristic is £42 + 2 = 10. If q, r be any half-integer
characteristics, it follows that any eleven functions of the form 32 (u, q) 32 (u, r) are
connected by a linear equation. Taking now, with Weierstrass's notation, the four
functions*
t = $&(u), x = 3M(u), ,y = 312(tt), z = \(u\
it follows that there exists an identical equation
in which the eleven coefficients A0, ...... , H2 are independent of u.
The characteristics of the theta functions #6(«), ^(u), 3l2(u), B0(u) may be taken,
respectively, to be (cf. § 220, Chap. XL)
/O, 0\ , /O, 0\ _ //>/, P2'\ . A 0\ _ /ft', ?2'\ /O, 0\ _ /<, rA .
U oj ' u i; - U.«; ' y ' Vi oj - u, & ; ' y ' U y ~ W; v ' y '
hence, by the formulae (§ 190)
5(M + 0P; ?) = eA^M)-2'riP'^(u; q + p),$(U't q + M } = <?***' 3 (u • q],
wherein M denotes a row of integers, we obtain
35 (U + Op) = «M«> ^34 («)> ^34 (« + 0P) = ^(W> -»6 («)> 512(« + Op) = <^(M) ^0 («),
hence the substitution of u + Qp for u in the identity replaces t, x, y, z respectively by
x, t, z, y. Comparing the new form with the original form we infer that
Similarly the substitution of u + Qq for u replaces t, x, y, z respectively by y, z, t, x ;
making this change, and then comparing the old form with the derived form, we infer
that
* Which are all even and such that the square of every other theta function is a linear
function of the squares of these functions. It can be proved that these functions are not
connected by any quadratic relation.
289] NUMBER OF SUCH RELATIONS. 467
Thus the identity is of the form
Taking now the three characteristics
(&', /A = /O, i\ /<?/, <72'\ _ ft, 0\ //V, h2'\ _ ft, £\
v/i, fj \o, oj ' u> ffj ~ vo, <>; ' U. v " v>, o; >
and adding to the argument u, in turn, the half-periods Q,, a,, OA and then putting u = 0,
we obtain the three equations
where 3* denotes ^(0), etc., and the notation is Weierstrass's, as in § 220. By these
equations the constants F, G, H are determined in terms of zero values of the theta
functions. The value of C can then be determined by putting u = Q in the identity
itself.
Thus we may regard the equation as known ; it coincides with that considered
in Exx. i. and iv. § 221, Chap. XL, and represents a quartic surface with sixteen nodes.
With the assumption of certain relations connecting the zero values of the theta functions,
proved by formulae occurring later (Chap. XVII. § 317, Ex. iv.), we can express the
coefficients in the equation in terms of the four constants S6(0), SM(0), 5ia(0), 40(0).
We have in fact, if these constants be respectively denoted by d, a, b, c
hence the identity under consideration can be put into the form
^-^+*^
where the ri denotes the product of the four factors obtained by giving to each of Cl, e2
both the values +1 and -1. The quartic surface represented by this equation (Ma 'be
immediately proved to have a node at each of the sixteen points which are obtainable
from the four,
(d, a, 6, c), (d, a, -b, -c), (d, -a, b, -c), (d, -a, -b, c),
by writing respectively, in place of d, a, b, c,
(i) (d, a, b, c), (ii) (a, d, c, 6), (iii) (b, c, d, a), (iv) (c, b, a, d).
Ex. iv. We have in Ex. iii. obtained a relation connecting the functions
in Ex. iv. § 221 we have obtained the corresponding relation connecting the functions
and in Ex. i. § 221 we have explained how to obtain the corresponding relation connecting
the functions
30—2
468 EXAMPLES. [289
There are* in fact sixty sets of four functions among which such a relation holds ; and
these sixty sets break up into fifteen lots each consisting of four sets of four functions,
such that in every lot all the sixteen theta functions occur, and such that in every lot one
of the sets of four consists wholly of even functions while each of the three other sets
consists of two odd functions and two even functions. This can be seen as follows : using
the letter notation for the sixteen functions, as in § 204, and the derived letter notation
for the fifteen ratios of which the denominator is 9 (u), as at the top of page 338, it is
immediately obvious, as on page 338, that any four ratios of the form
l> 2*,*. 2Al.V 2*, »
in which the letters k, I, £lf llt k% constitute in some order the letters alt «2, c> ci> C2> are
connected by a relation of the form in question. Now such a set of four ratios can be
formed in fifteen ways ; there are firstly six such sets in which all the ratios are even
functions of u, obtainable from the set
1> 2e» 2a,, c,» ?aa, c3
by permuting the three letters c, clt c2 among themselves in all possible ways ; and nextly
nine such sets in which two of the ratios are odd functions, obtainable from the set
by taking instead of the pair a^ each of the three pairs t a1«2, aalt aa2, and instead of
the pair CjC2 each of the three pairs c^, ccj, cc2. Since (§ 204) the letter notation for an
odd function consists always of two a-s or two e-s, and for an even function consists of
one a and one c, the number of odd and even functions will remain unaltered. Further
from each of these fifteen sets we can obtain three other sets of four ratios by the addition
of half-periods to the argument u, in such a way that all the sixteen theta functions
enter into each lot of sets. The fifteen lots obtained may all be represented by
the scheme
1, a , /3 , a/3
«i > aai > ^ai > a/3ai
ft , aft , #3, , a/3ft
ajft, aajft, /Sajft, a/3aift,
where 1, a, ft a/3 denote the characteristics of one of the fifteen sets of four theta functions
just described, such as S (u), &e(u), \Ci (w), ^ (u), or &(u), $c(u), \tta («), ^(u),
aft denoting the characteristic formed by the addition of the characteristics a, /3 ; and at , ft
denote any other two characteristics other than a, ft or aft and such that a/3 is not the
same characteristic as axft. This scheme must contain all the sixteen theta functions ;
for any repetition (such as a = /3ajft, for example) would be inconsistent with the
hypothesis as to the choice of a,, ft (would be eqxiivalent to a^ = ajft). It is easily seen,
by writing down a representative of the six schemes in which the first row consists
wholly of even functions, and a representative of the nine schemes in which the first
row contains two odd functions, that in every scheme there are three rows in which two
odd functions occur J.
Ex. v. There are cases in which the number of linearly connected theta functions, as
given by the general theorem, is subject to further reduction. For instance, suppose we
* Borchardt, Crelle, LXXXIII. (1877), p. 237. Each of the sixty sets of four functions may be
called a Gopel tetrad.
t The letter a, when it occurs in a suffix, is omitted.
J A table of the sixty sets of four theta functions is given by Krause, Hyperelliptische
Functionen (Leipzig, 1886), p. 27.
289] REDUCTION OF THE CHARACTERISTIC. 469
have w = 2p~1 odd half-integer characteristics Alt ..., Jm, and another half-integer charac
teristic P, not (integral or) zero, such that the characteristics* A{P, ..., AmP, obtained
by adding P to each of Alt ..., Am, are also odd + ; suppose further that A is an even
half-integer characteristic, and that ylPis also an even characteristic, and that the theta
functions 3 (u ; A), 3(u; AP) do not vanish for zero values of the argument. Then, by
§ 288 the W^ + l following theta functions of order 2,
3(u; A)3(u; AP), 3 (u • Al)3(u; A,P), ..., 3(u; Am)3(u; AmP),
which are all even functions with a characteristic differing only by integers from the
characteristic P, are connected by a linear equation with coefficients independent of u.
But in fact, if we put u = 0, all these functions vanish except the first. Hence we infer
that the coefficient of the first function is zero, and that in fact the other 2p~l functions are
themselves connected by a linear equation.
Ex. vi. In illustration of the case considered in Ex. v. we take the following : — When
/> = 3, it is possible £, if P be any characteristic whatever, to determine six odd characteristics
A19 ..., A6, whose sum is zero, such that the characteristics AtP, ...,A6P&re also odd, and
such that all the combinations of three of these, denoted by AiAjAk, AiAjAkP, are even.
By the previous example there exists an equation
AS (u ; J4) 3 (u ; J4P)
; A2)3(u; A2P)+\33(u; As) 3 (u • A3P),
wherein X, A1} A2, A3 are independent of u. Adding to u any half-period Qfi, this equation
becomes
A3 (u ; AiQ) 3 (u ; AtPQ)
= X1«1S(M; AlQ)S(u; A1PQ) + \2f23(u; A2Q)3(u; AzP<j) + \^(u; A3Q)3(u; A3PQ),
where fi(i=l, 2, 3) is a certain square root of unity depending on the characteristics
^4> ^i> P, Q, whose value is determined in the following example. Taking in particular
for Q2 the half-period associated with the characteristic AZA3, so that the characteristics
A2PQ, A3PQ become respectively the odd characteristics A3P, A2P, and putting « = 0,
we infer
A4(0; A4AZA3)3(0- A,A2A3P) = \lfl'3(0 • ^2J3)3(0; A,A2A3P),
where */ is the particular value of fl when Q is A2A3. This equation determines the ratio
of At to A ; similarly the ratios A2 : A and A3 : A are determinable.
Ex. vii. If £r, $q be half-integer characteristics whose elements are either 0 or £, and
$k=$rq be their reduced sum, with elements either 0 or £, prove § that
*. = »-. + ?.-2rB?a> *.' = »-.' + ?.'- 2ra'?a', (a = l,2, ...,p),
and thence, by the formulae (§ 190)
e**™* S (u ; 2),
* A characteristic formed by adding two characteristics A, P is denoted by A + P. Its
reduced value, in which each of its elements is 0 or £, is denoted by AP.
t It is proved in chapter XVII. that, when j»2, the characteristic P may be arbitrarily
taken, and the characteristics Alt...,Am thence determined in a finite number of ways.
t This is proved in chapter XVII.
§ Schottky, Crelle, en. (1888), pp. 308, 318.
EXAMPLES. [289
where M is integral, prove that
X (u ; i r\ + wi I (raga?a' + gjy-J
3 (w+i QP ;£?) = «
If ^ ^a, \q be any reduced characteristics, infer that
where
«r 2 [ragfta.' + (raga' + ra'?a + ra') o J
e = e a=1 -; .;
^p. viii. If ^u ^42, ^3, AI denote four odd characteristics, for jt> = 2, and 5 denote an
even characteristic, the 12» + 2P-1 + 1 = 5 theta functions, of order 2 and zero (or integral)
characteristic, &(u; B), &(u; AJ, ...,52(«; Aj are, by § 288, connected by a linear
equation. As in Ex. v. we hence infer an equation of the form
adding to w the half-period associated with the characteristic AZA3, and putting M=0, we
deduce by Ex. vii. that
Xe**1'0* 52 (0 ; A,AzA.} = \,eivk^ V (0 ; ^^2^3),
where A2Az = \k^ Al = ^a1, ^4 = ^a4. Hence we obtain an equation which we may write
in the form
where (A^At3\ denotes a certain square root of unity. Such a relation holds between every
VM4/
four of the odd theta functions.
If A !,..., A6 be the odd characteristics, and Q be any other characteristic, the six
characteristics A$t ..., A6Q are said to form a Kosenhain hexad. It follows that the
squares of every four theta functions of the same hexad are connected by a linear relation.
291]
CHAPTER XVI.
A DIRECT METHOD OF OBTAINING THE EQUATIONS CONNECTING ^-PRODUCTS.
290. THE result given as Ex. xi. of § 286, in the last chapter, is a
particular case of certain equations which may be obtained by actually
multiplying together the theta series and arranging the product in a
different way. We give in this chapter three examples of this method, of
which the last includes the most general case possible. The first two furnish
an introduction to the method and are useful for comparison with the
general theorem. The theorems of this chapter do not require the charac
teristics to be half-integers.
291. Lemma. If b be a symmetrical matrix of p2 elements, U, V, u, v,
A, B, f, g, q, r,f, g', q', r, M, N, s, t', m, n be columns, each of p elements,
subject to the equations
n -f m = 2N + *', q' + r'=f', q + r=f,
then
2 U (n + q) + b (n + qj + Zniq (n + q') + 2 V (m + r') + b (m + rj + 2-Trir (m + r')
±/' + 26
This the reader can easily verify.
Suppose now that the elements of s and t' are each either 0 or 1, and
that n and m take, independently, all possible positive and negative integer
values. To any pair of values, the equations n + m = 2JV + s', — n + m = 2M + 1'
give a corresponding pair of values for integers N and M, and a pair of
values for s' and t'. Since 2m = 2JV+ 2M + s' + t', s' + t' is even, and there
fore, since each element of s' and t' is < 2, s' must be equal to t'. Hence by
means of the 2? possible values for s', the pairs (n, m} are divisible into 2^
sets, each characterised by a certain value of s'. Conversely to any assignable
472 ACTUAL CALCULATION OF PRODUCT [291
integer value for each of the pair (N, M) and any assigned value of s (< 2)
corresponds by the equations n = N—M, m = N+M + s' a definite pair of
integer columns n, m.
Hence, b being such a matrix that, for real x, ba? has its real part negative,
(n+tf) +b (n+q1) « +2mq (n+q') j T^g2 V (m+r') +b (m+r1) s+2jrir (m+r') 1
m
-z
thus, if *(u; \), or *u; , denote S^ujn+vj+Kn+vj^+shrtxjn+x^ ^(Mj X) or
V A,/ n
/ A/\
^ ft*; denote 2e4M(n+V)+26(n+v>2+2'riA(n+v>, we have
V A/ n
_
where the equation on the right contains 2? terms corresponding to all
values of s', which is a column of p integers each either 0 or 1 ; all other
quantities involved are quite unrestricted.
Therefore if a be a symmetrical matrix of p* elements and h any matrix
of p- elements, we deduce, replacing u by hu, and v by hv, and multiplying
both sides by eau'+av\ the result
where e denotes all possible 2^ columns of p elements, each either 0 or 1,
and ^ differs from S- only by having 2a, 2h, 2b instead of a, h, b in the
exponent ; thus we may write, more fully,
2&),
jEvp. i. When the characteristics q, r are equal half-integer characteristics, say
>', ""i '-
the equation is
multiplying this equation by ewwn, when n denotes a definite row of integers, each either
291] OF TWO THETA FUNCTIONS. 473
0 or 1, and adding the equations obtained by ascribing to a all^he 2^' possible sets of values
in which each element of a is either 0 or 1, we obtain
for we have
a f=l
Ex. ii. Deduce from Ex. i. that when p = l, the ratio of the two functions
is independent of «..
^r. iii. Prove that the 2?> functions ^ ( u ; ft ) j obtained by varying «', are not
connected by any linear equation with coefficients independent of u.
Ex. iv. Prove that if a, a' be integral,
From this set of equations we can obtain the linear relation connecting the squares of
(or less) assigned theta functions with half-integer coefficients.
Ex. v. Using the notation (X,-,,-) for the matrix in which the j-th element of the i-th
row is Xf,y, prove that if ul, ..., u,., vt, ..., vr be 2.2? arguments, and $(a J any half-
integer characteristic,
.[•>' *o]
and, denoting the determinant of the matrix on the left hand by {?/f, v}} and the determi
nant of the second matrix on the right hand by {v}, deduce that
where A is the sum of the p elements of the row letter a. When the characteristic | ( J
is odd, {u(, Uj} is a skew symmetrical determinant whose square root is* expressible
tionally in terms of the constituents $Ffc|+1^j i{* ji *[««<-% i iff]- For
instance when p = I, we obtain, with a proper sign for the square root, the equation of
three terms t.
Since any 2* + 1 functions of the form 3 \ u+vp ; \ ( a J 5 1 u - vp ; % ( a j | are connected
by a linear equation with coefficients independent of u, it follows that if w1? ..., um,
vi> ...» ^m be any 2m arguments, m being greater than 2", the determinant of m rows and
columns, whose (/, »th element is *|n+«y; ^ (°)~|^ F^- vs ; i(")l, vanishes identi
cally. When ^ f a j is odd and m is even, for example equal to 2" + 2, this determinant is
* Scott, Theory nf determinants (Cambridge, 1880), p. 71.
t Halphen, Fonet. EUip. (Paris, 1886), t. I. p. 187.
474 CALCULATION OF THE PRODUCT [291
a skew symmetrical determinant whose square root may be expressed rationally in terms of
the functions 5| tlf-HV j ill rH ui~vi > if) • The resu^ obtained may be written
{ut, «,-}* = (),
wherein* the determinant {uit v}] has m rows and columns, m being even and greater than
2 p. When m is odd the determinant {«;, Vj] itself vanishes.
A proof that for general values of the arguments the corresponding determinant
{ut, ^j}, of 2P rows and columns, does not identically vanish is given by Frobenius, Crelle,
xcvi. (1884), p. 102.
A more general formula for the product of two theta functions is given below
Ex. ii. § 292.
292. We proceed now to another formula, for the product of four theta
functions. Let J denote the substitution
l 1 1 1),
1-111
11-11
111-1
and Jrs be the element of the matrix which is in the r-th row and the 5-th
4
column ; then 2 Jir Jis = 0 or 1, according as r =f= s, or r = s (r, s = 1, 2, 3, 4).
i = l
Let MU u2, u3, u4 denote four columns, each of p quantities; written down
together they will form a matrix of 4 columns and p rows. Let U1} U2, U3,
U4 be four other such columns, such that the j-th row of the first matrix
(j = 1, 2, ..., p) is associated with thej-th row of the second by the equation
Let vlt vz, v3, v4 and V1} F2, Vs, V4 be two other similarly associated sets,
each of four columns of p elements. Then if h be any matrix whatever, of p
rows and columns, we have
hu^ + h uzvz + hu3v3 + hu4v4 = hU1V1 + hU2V2 + hU3V3 + hUtV^;
this is quite easy to prove : an elementary direct verification is obtained by
selecting on the left the term hj^u^v^j + hjk(u2)k(vz)j+hjk(u3)k(v3)j+ hj
= hjk 2 [Jrl ( ujk + J* ( u,\ + Jr3 ( u3)k + Jr4 ( u4)k] [Jrl
= hk
= hjk {(U1\(Vl)j + (U2)k (F2)j
and this is the corresponding element of hUl Vl + hU2V2 + hU3V3 +hU4V4.
* The theorem was given by Weierstrass, Sitzungsber. der Berlin. Ak. 1882 (i. — xxvi., p. 506),
with the suggestion that the theory of the theta functions may be a priori deducible therefrom, as
is the case when|? = l (Halphen, Fonct. Ellip. (Paris (1886)), t. i. p. 188). See also Caspary,
Crelle, xcvi. (1884), and ibid. xcvu. (1884), and Frobenius, Crelle, xcvi. (1884), pp. 101, 103.
292] OF FOUR THETA FUNCTIONS. 475
Now we have
Sr (Ml, ?1) ^ (u.2, qa) $• (u3, q3}
In the exponent here there are four sets each of four columns of p quantities
namely the sets
Mr, Kr, qr, tfr',
we suppose each of these transformed by the substitution J. Hence the
exponent becomes
Flf 2T,, 2T,. JT,
wherein the summation extends to all values of Nrj given by
t
Nrj = % (nl} + nzj + n3j + n4j - 2nrj),
for which all of nrj are integers.
All the values Nrj will not be integral. But since Nrj — Ngj = rigj — nrj the
fractional parts of N^, N2j, N3j, Ntj will be the same, = £ e/, say, (e/ = 0 or 1).
Let rrirj be the integral part of Nrj. We arrange the terms of the right hand
into %P classes according to the 2? values of e/. Then since
every term of the left-hand product, arising from a certain set of values of
the 4>p integers nrj, gives rise to a definite term of the transformed product on
the right with a definite value for e/, while, since
every assignable set of values of the 4<p integers mrj and value for e/ (which
would correspond to a definite term of the transformed product) will arise,
from a certain term on the right, provided only the values assigned for mrj be
such that ^ (rn^j + m%j + m3j + m4j + e/) is integral.
Now we can specify an expression involving the quantities
A*j, =1 0»ij + m*j + msj + m4j + c/),
which is 1 or 0 according as ft = (/A1, /A2, ..., fip) is a column of integers or
not. In fact if e = (e!, ..., ep) be a column of quantities each either 0 or 1 —
so that e is capable of 2^ values — the expression
J_ 2e*nV = 1 (^ea-nv.M,) . . . (^ez™^ = J_ (i + e2«>,) (j + ezm^ . . . (i + #**,)
has this property ; for when ^ , . . . , /j.p are not integers they are half-
integers.
476 FOUR THETA FUNCTIONS. [292
Hence if the series =- 2e7ri'6(mi+"l2+m3+m<+O be attached as factor to every
2P f
term of the transformed product on the right we may suppose the summation
to extend to all integral values of mrj,for every value of e.
Then the transformed product is
1
V pZaUr+VZhUr (mr+be'+Q'rl+'S.b (mr+K+Q'r)2+2Jrt2Qr (mr+Je' + QVl+jrie (?«,+Jn1!+?ft3+m1+e')
Op
sf mlm.im3m,te
1 2
= _ ^n^Ur+'ihUr (mr+p'r)+b (mr+p'r) *+2iripr (mr+p'r) _ Q~^f (2pV-«'>
2P r
where
Pr=^€+Qr, Pr=^€+Qr,
so that
2p/=2e' + 2Q/=2e' + %'.
Thus we have
S- (MJ, g,) ^ (u2, qj % (u3, q3) ^ (w4, q4)
'^
This very general formula obviously includes the formula of Ex. xi., § 286,
Chap. XV. It is clear moreover that a similar investigation can be made for
the product of any number, k, of theta-functions, provided only we know of a
matrix J, of k rows and columns, which will transform the exponent of the
general term of the product into the exponent of the general term of the sum
of other products.
It is for this more general case that the next Article is elaborated.
It is not necessary for either case that the characteristics q1} q2, ... should
consist of half-integers.
Ex. i. If q be a half- integer characteristic, = Q, say, and we use the abbreviation
0(M, v, w, t\ Q) = $(u; Q)3(v, 0)3(1*', 0)*(t> Q},
we have
i
(f)(u + a, u-a, v + b, v-b, Q) = ^ 2 e~nife' <f>[u + b, u-b, v + a, v-a ;
• f,f'
where the summation on the right hand extends to all possible 22p half-integer character
istics £ r J ; putting Q + $(f\ = R, so that R also becomes all 22^ half-integer character
istics, this is the same as
, u-a, v + b, v-b; Q) = >2e"ilQ' Rl + wi [Rl <f>(u + b, u-b, v + a, v-a; R),
where,
if <?
293]
GENERAL CASE.
477
By adding, or subtracting, to this the formula derived from it by interchange of v and
a, we obtain a formula in which only even or odd characteristics R occur on the right hand.
Thus, for p = l, we derive the equation of three terms.
Ex. ii. If a, |3, y, 8 be integers such that ay is positive and /38 is negative, p = a8-j3y,
and r be the absolute value of p, prove that
ayr
/38r
e -
where e f w ; r J denotes the theta function in which the exponent of the general term is
2 iriu (n + e') + iirr (n + 1 ')2 + 2irie (n + e'),
and |t, «* are row letters of p elements, all positive (or zero) and less than r, subject to the
condition that (8/x - /3i>)/p, (av — y/i)/p are integral, while e, /, g, h are row letters of p
elements which are all positive (or zero) and less than r.
Ex. iii. Taking, in Ex. ii., a, @, y, 8 respectively equal to 1, 1, 1, -k, we find
p = v<k+ 1, k being positive. Hence, taking & = 3, prove the formula (Konigsberger,
Crelle, LXIV. (1865), p. 24), of which each side contains 2P terms,
26
(u; r\\S>
V It
QU; 3r I f $'-
if
's 9 /O; r °.} Q (lu; 3r ! ? V
\ liv V If*/
s, «' being rows of p quantities each either 0 or 1.
293. We proceed now to obtain a formula* for the product of any
number, k, of theta functions.
We shall be concerned with two matrices X, x, each of p rows and k
columns ; the original matrix, written with capital letters, is to be trans
formed into the new matrix by a substitution different for each of the
p rows ; for the j-th row this substitution is of the form
(Xij, X2j, ..., Xrj, ..., Xktj)=- a>j(xliit asaj, ..., xrj, ..., ocktj);
rj
herein TJ is a positive integer; to,- is a matrix of k rows and columns,
consisting of integers ; the determinant formed by the elements of this
matrix is supposed other than zero, and denoted by /^; bearing in rnind
that throughout this Article the values of r are 1, 2, ..., k and the values of^'
are 1, 2, . . . , p, we may write the substitution in the form
The substitution formed with the first minors of the determinant of etj will
be denoted by ft,-; that formed from flj by a transposition of its rows and
columns will be denoted by IL. Then the substitution inverse to - «, is
ri
£fy'} denoting the former substitution by X,-, the latter is X/-1.
Prym und Krazer, Neue Grundlagen...der allgemeinen thetafunctionen, Leipzig, 1892.
478 INVESTIGATION OF A GENERAL FORMULA [293
If for any value of j a set of k integers, Prj, be known such that the k
quantities
are integers, then it is clear that an infinite number of such sets can be
derived; we have only to increase the integers Prj by integral multiples of
jj,j. But the number of such sets in which each of Prj is positive (including
zero) and less than the absolute value of ft is clearly finite, since each
element has only a finite number of possible values. We shall denote this
number by Sj and call it the number of normal solutions of the conditions
A*. _
— flj (Pr,j) = integral ;
ft
it is the same as the number of sets of k integers, positive (or zero) and less
than the absolute value of ft, which can be represented in the form ^j(pr,j),
for integral values of the elements prj.
The k theta functions to be multiplied together are at first taken to be
those given by
®r = 2e2 W+^W (r = 1 ,...,&),
wherein Br is such a symmetrical matrix that, for real values of the p
quantities X, the real part of the quadratic form denoted (§ 174, Chap. X.) by
BrX2 is negative. The p elements of the row-letters Vr, Nr are denoted by
Vrj, Nr>j(j = 1, ...,p). The substitutions Xj are supposed to be such that
k
the equations (Xr> j) = *hy(ar,j) transform the sum 2 BrXr2 into a sum
r=l
t
S brXr*> in which the matrices br are symmetrical and have the property that
r=l
for real xr the real part of byX? is negative.
Taking now quantities mrj, vrj determined by
(mr, j) = X,--1 (Nr> j) = p n,- (Nr> j), (vr, j) = \j ( Vr> j) = - &j ( Vr, j),
/*; rj
k k k
the expressions X BrNrz, S NrVr are respectively transformed to 2 brmt?
r=l r=l r=l
and
p P —
2, \j (mr> j) ( Vr> j) = S Xj ( Vr> j) (mr, j) = 2 vrmr ;
j=l j=l r=l
hence the product II ®r is transformed into 1 e%"rWr , rmr t where the
r=l N, ..... Nk
quantities mrj have every set of values such that the quantities \j(mrj) take
all the integral values, Nrj, of the original product.
293] FOR THE PRODUCT OF THETA FUNCTIONS. 479
As in the two cases previously considered in this chapter, we seek now to
associate integers with the quantities mrj. Let (Pr,j) be any normal solution
of the conditions
^ tij (Prj) = integral, = (pr< j), say ;
Pt
put, for every value of j,
(Nrj)-(Prj) = Hj(Mrtj) + (E'rJ\ (r = l, ..., k)
wherein (Mrj} consists of integers, and (E'r,j) consists of positive integers
(including zero), of which each is less than the absolute value of /j,j. For an
assigned set (Pr> j) this is possible in one way ; then
(mr, j) = 2 Qj(Nrti) = (pflj) + rjQj (MrJ) + ^ ^(E'r>j}
Pi Pi
= (nr,j)+ -;(e'r,j), say,
where
(«r,j) = (Pr.j) + rjflj (MrJ), (e'r>j) = Tj&j (E'rj) '
by this means there is associated with (Nrj), corresponding to an assigned
set (Pr,j), a definite set of integers (nrj), and a definite set (E'rj). We do
not thus obtain every possible set of integers for (nrj), for we have
~ «/ K, j) = £ «j (Pr, j) + ft (Mr, j) = (Pr, j) + HJ (Mr> j),
Tj Tj
so that the values of nr> j which arise are such that Xj (wr> j) are integers.
Conversely let (nr. j) be any assigned integers such that \j(nrj) are
integers ; put
\j(nr>j) = (PrJ) + ftj(Mr!J),
wherein the quantities Mrj are integers, and the quantities Prj are positive
integers (or zero), which are all less than the absolute value of /^-; this is
possible in one way ; then taking any set of assigned integers (E'r, j), which
are all positive (or zero) and less than the absolute value of fij, we can define
a set of integers Nr< j by the equations, wherein \f~l (Pr> j) = integral,
(Nrtj) = (E'rJ) + (PrJ) + pj (MrJ) - (E'r>j) + X,. (nrj).
Thus, from any set of integers (Nrj), arising with a term e*(Z rNr+SrNr*) of
I
the product II ®r, we can, by association with a definite normal solution
r=l
(Pr,j) of the conditions \f* (Prj) = integral, obtain a definite set (E'rij), and
a definite set (nr,j) such that \j(nrj) are integers. And conversely, from any
set of integers (wr,j) which are such that \j(nfij) are integral, we can, by
association with a definite set (E'r>j), obtain a definite normal solution (Pr,j)
and a definite set (Nrj).
480 INVESTIGATION OF A GENERAL FORMULA [293
It follows therefore that if the product H ®r be written down ^ . . . sp times,
r=l
a term er being associated in turn with every one of the Si ...sp
normal solutions of the p conditions X/"1 (Pj) = integral, then there will arise,
once with every assigned set (E'Tj j), every possible set (nr> j) for which Xj (nr< j)
are integers.
We introduce now a factor which has the value 1 or 0 according as the
integers (nTi j) satisfy the conditions X,- (nri j) = integral, or not. Take k in
tegers (Erj), which are positive (or zero), and less than r,-; put
then
. j_ !?jj ] = 22 — e, ,- mr 4 = 2X9- (Er ,•) (mr ,-) = 2X,- (mr ,•) (Er ,-)
>j I „-_ ' i j 'ij . j \ >tj/ \ M^/ . j \ <j' » 'j'
j
and this is integral when Nr is integral, that is, for all the values (nrj) which
actually occur ; in fact the quantities Nr> j denned by
1 / ' \ ~\
(Nr>j) = \j(mrt}) = - ^(%^-t-^) = - «!/ («r, j) + (E'rj) = ^ Kj) + (E'rJ)
rj \ n ' 'i
are integral or not according as Xj (nfj j) are integers or not.
Hence, for a given set nrjy and a given set E'rj, the sum
wherein the summation extends to all positive (and zero) integer values of
(Erj) less than TJ, is equal to r^ ... r^j when (Nrj) are all integral, and other
wise contains a factor of the form
which is zero because TJ (Nr,j) is certainly integral. Hence if we denote
22 - erj (nri j + e^} by S I er f n, + ^) ,
j r r,- 3 V /A? / r ^ V /* /
J^ having the values r^ ..., rp, then we can write
1 v ZTriZ^er (nr+-\ n n
, - ^2e r« \ ft/ = l, or 0,
(n . . . r^f .E
according as X^ (?ir> j) are all integers or not.
If then every term of the transformed series, in which, so far, only those
values of nrj arise for which \j(nrj) are integers, be multiplied by this factor,
293] FOR THE PRODUCT OF THETA FUNCTIONS. 481
and the transformed series be completed by the introduction of terms of the
same general form as those which naturally arise in this way, so that now all
possible integer values of (nr>j) are taken in, the value of the transformed
series will be unaltered. In other words we have
n,E,E r
wherein all possible integer values of (nrj) arise on the right ; thus the right-
hand side is equal to
E\ E r
and this is the desired form of the transformed product. For con
venience we recapitulate the notations; Er', Er each denote a column of
p integers, positive or zero, such that E'r>j < ^ , ErJ < r,- ; (Yr,j) = rjfy (E'ft})\
(er>j) = ajj(E,.j); Sj is the number of sets of integral solutions, positive or
zero, each less than ^ , of the conditions ij.j~lrj£Lj(Prt ,-) = integi-al ;
(vr, j) = rj~l 6>j (Vr> j) ; the function ®,. is a theta function in which the ordinary
matrices a, b, h (§ 189) are respectively 0, 6,., 1 ; by linear transformation of
the variables of the form Vr = hrWr, and, in case the matrices coj be suitable,
•S.ArV^
multiplication by an exponential er , these particularities in the form of
the theta functions may be removed.
The number of sets (Erj) is (^...rp)*; the number of sets (E'rtj) is
l/Ltj* ... npk ; the product of these numbers is the number of theta-products on
the right-hand side of the equation.
Ex. i. We test this formula by applying it to the case already discussed where «,• is
an orthogonal substitution given by
which is independent ofj, r,-=2, 6r =
= 4; then ^= - 16, -Er,j<2, E'rJ<lG, and
thence -j\(i,j- ^f2,y=-^2,>~-^u = integral, etc., so that the fractional part of j,tr,j is in
dependent of r : similarly the fractional part of - (e'r>>) is independent of r and we may
write (f'r.j) = (kf'j + I'i.j, \t'j + L2<i, ..., ^t'j + L^) wherein 2ZriJ- + e'y<16. By the formula
B. 31
482 EXAMPLES OF THE APPLICATION [293
= eMq>ff S(v, q}, when N is integral, we know that 9,. ( v,. ; *][£] is independent
\ fr/-*V
of the integral part of f'r/p. Hence the (16)4» = 216') terms on the right-hand side of the
general formula, which, for a specified value of £» (Er>i\ correspond to all the values of
\a>(E'r,j\ reduce to 2P terms, in which, since (E'r,j) = i<» (a f'i + ^i,i > •••> ^6'j + A,j)> *U
values of e'(<2) arise. Hence there is a factor 216p and instead of the summation in
regard to E, E' we have a summation in regard to E, e', the right hand being in fact
(7.2«p 2 neftv, f6'
A', e' »• \ |« (,A
and containing 24j> terms.
Now put $(Ei,j+E2,i + E3,s + E4,J
Mj being integral ; then the factor of a general term of the expanded right-hand product
which contains the quantities %a> (Eftj) is
ne2iriifcr (»,.+*£')
}•
where
kr,J = E1,j + E,,j + E3,j + Et,j-2Er,j = (
and
farie"SJcr — 11$™)' (tej+SMj—Zej-lMj) _ jj
* i 3 '
while
22Trik,.tjnrtj = iri22fjnrj (mod. 2), =iric . 2»r,
j r j r r
so that
ne27riP,.(n,.+K) _rne27r^€(nr+Je')-j g-««'.
,. ,.
therefore the right-hand product consists only of terms of the form 116 (v,., f J e~"M£.
Hence the 24^ terms arising, for a specified value of e', for all the values of Erti) reduce to
2p terms, and there is a further factor 23j) — the right hand being
where
C7-(ii...^)-1(rI...rf)-*-(ik-i)"1»"¥-»"*1"IPli
To determine the value of C we must know the number (s) of positive integral
solutions, each less than 16, of the conditions £o> (#) = integral, =(y) say, namely of the
conditions, x1 + x2 + x^+xi = 2(xr + i/r}. Now of these any positive values of x^ x^ xz, x±
(<16) are admissible for which ^ + ^2 + ^3+^4 is even. They must therefore either be
all even, possible in 84 ways, or two even, possible in 6 . 82 . 82 ways, or all odd, possible in
84 ways. Hence s = 8 . 8^ = 2^. Hence <7=1/215"24P = 1/219P and therefore C. ^8p = ^-
Making now in the formula thus obtained, which is
the substitution Vr = kUr, we have vr = $(Vl + Vs+V3+ F4 - 2 T"r) = hur, where
Ur=%(U1+U2+U3+Ui-2Ur); and if we multiply the left hand },;}
which is equal to e^+^+^+^\ we obtain
293] OF THE GENERAL FORMULA. 483
Therefore if Qly Q2, Q3, Q^ denote any characteristics, and, as formerly, QQ denote the
period-part corresponding to Qr, we have
n3(Ur, Qr) = ne-^ur> ^»(Ur+QQ o)=n<rA<^' «-) n*(Ur+aQ , o),
of which the first factor is easily shewn to be mT^' *•>, if fa, ?2, ?3, ?4) = ^w(<21, &, Q3, Qt) •
thus
, Qr)= e-™'ne-^<r> *)
which is exactly the formula previously obtained (§ 292).
Ex. ii. More generally let A = -<o,- be any matrix such that the linear equations
wherein m is independent of ^, ..., xk ; then, since, by a property of all linear substitutions,
the equations (Yr) = \ (yr) lead to
r a ^v 8 3 9
Jla^ + ...... + F^ryi8^+ ...... +^9V
we have also*
Hence, if A be any matrix of p rows and columns and
(Xrti') = \(xrti\ O'=l, ...,P),
we have
^2^,. Fr> < = j» 2 ^i,^.
r i,jr
where Ar1} a,\, etc. now denote rows of p quantities.
tution furnishes a
,- = «, n= +r*, ^r>y < r, A"P>>
Thus any orthogonal substitution furnishes a case of our theorem. Taking a case
where
we have
so that the new characteristics will be r-th parts of integers.
Suppose now, in particular, that the substitution is
*
2 2-/t... 2
2
* Therefore mxy = XY=\x . \y = \\xy, so that \\=m ; hence the determinant formed with the
elements of X has one of the values ^/m*.
31—2
484 EXAMPLES OF THE APPLICATION [293
which gives
= 2 R ; Oi + . . . + #fc) - xr = k ^
. . . . . . . . . + xk)'2 + 2A>2 - ^ 2^,. (^ + . . . + xk) = 2 J?r8
and
A\ + ...... +Xk = A\ + ...... + A'fc, Ar1-Z2 = ^2-^1, etc.
The previous example is a particular case, namely when £ = 4. In what follows we
may suppose k odd so that r,- = &. When £ is even r, may be taken = £&. The work is
arranged to apply to either case.
The fractional parts of - («',., y) being independent of the suffix r — because
e\,j--<'-2,J = -E'>t,J-E'i,i> etc->
—we may put -(?',-, j) = (-,«/+ A, >> •••> - fj' + ^k,j] > aild may therefore write
ne f *v 'r!£\ in the form ne (v,., ' JV) .
r \ frltij r \ frl^-J
The equation
shews that all values of - e/ (< 1) do arise. Hence for a given value of (Efti) there are,
instead of 1^1^ = ?-^' terms given by the general formula, only r'\ and the factor H*'2"1^
divides out.
The values of —(*,-,,•) given by the general formula are in number r **>, corresponding
to all the values of (Er>j). As before the fractional part of - («,., ,-) is independent of r. Let
where T<! ; then
K
,., .
The factor in the general term of the expanded product on the right hand which
contains cr>j is
Now
1
2ri / 77f \
— €,. ,- = 2 (Jiir j) =<
r 1' r
therefore, as r is k or a factor of k,
Zm — e • — e' Znite + IcM •) ei Ztri'j^i-
Ile r'rj=e } } r —e f
r
and
= 2 ft + 2 J/,- - /?r , f] nr , ,- = -k ^fjnr , ,. (mod. 1 ).
293]
OF THE GENERAL FORMULA.
485
Hence the factor above is
r 2e / e\~\ te' «'
T- ^ 2lrl'T (" + -) -Jrt— 2ni —
K=\l\.e k \ r r/ \e >• e r ,
Lr
and the general term of the right hand is
[••(*• £)>•"*•
1 /2f, \
06 7?(fr>*' I ir+^J^ — ETti ] we may suppose all values of tj<* to arise. Hence
\ K j
instead of I*P we have k*> and a factor i*i>\ki> divides out.
To evaluate the factor (^ ... rp}-1 (sl ...s]t)-k, =C, say, we must enquire how many
positive solutions exist of the conditions
-#,. = integral,
£ (#1 + ......
namely, how many solutions of the conditions
2
T (-^i 4- +%k) = integral,
exist, for which each of .rt, ..., xk<rk ; let s be this number ; then C=s-Pr~kJ), and
ne(Vr,0) = ^~^- 5
where «'<r, e < ^r, the number of terms on the right being (rky. For values of f> - we
may utilise the equation 3(v, q + N) = e~niNq' 3 (v, q). For example, when * = r = 3 there
are 3* terms, corresponding to characteristics (*/? ) . When k = 4, r=2, the character-
2e e
istics -£ = - will, effectively, repeat themselves. We can reduce the number of terms from
8» or 23P to W. We shall thus get factors
that already found.
= l and so the formula reduces to
Ex. \\i. Apply the formula of the last example to the orthogonal case given by «, = w,
(A', Y, Z, T, U, K) = £«(#, y, z, t, u, v),
<•> = ( 1 1 0 0 1 -1 ), «-i = <
1 1 00-1 1
0 0
0 0
1-1 1 1
-1111
001-111
00-1 1 1 1
which lead to = 64 and
1 1 1-1 0 0 ),
1 1-1 1 00
001 1 1-1
0011-11
1-1 00 1 1
-110011
A' + r +Z +T + U + V =.v +y +z +t +u
Z-T=x-y, U-V=z-t, X-Y=u-v,
[294
CHAPTER XVII.
THETA RELATIONS ASSOCIATED WITH CERTAIN GROUPS OF CHARACTERISTICS.
294. FOR the theta relations now to be considered*, the theory of the
groups of characteristics upon which they are founded, is a necessary
preliminary. This theory is therefore developed at some length. When the
contrary is not expressly stated the characteristics considered in this
chapter are half-integer characteristics f ; a characteristic
i
Wi, q-2, -.., qp
is denoted by a single capital letter, say Q. The characteristic of which all
the elements are zero is denoted simply by 0. If R denote another charac
teristic of half-integers, the symbol Q + R denotes the characteristic, 8 = |s,
* The present chapter follows the papers of Frobenius, Crelle, LXXXIX. (1880), p. 185, Crelle,
xcvi. (1884), p. 81. The case of characteristics consisting of rt-th parts of integers is considered
by Braunmiihl, Math. Annal. xxxvn. (1890), p. 61 (and Math. Annal. xxxn. (1888), where the
case ?i = 3 is under consideration).
To the literature dealing with theta relations the following references may be given : Prym,
Untersuchungen iiber die Riemanri'sche Thetaformel (Leipzig, 1882) ; Prym u. Krazer, Acta Math.
in. (1883) ; Krazer, Math. Annal. xxn. (1883) ; Prym u. Krazer, Neue Grundlagen einer Theorie
der allgemeinen Thetafunctionen (Leipzig, 1892), where the method, explained in the previous
chapter, of multiplying together the theta series, is fundamental : Noether, Math. Annal. xiv.
(1879), xvi. (1880), where groups of half-integer characteristics are considered, the former paper
dealing with the case p = i, the latter with any value of p; Caspary, Crelle, xciv. (1883), xcvi.
(1884), xcvii. (1884) ; Stahl, Crelle, LXXXVIII. (1879) ; Poincare", Liouville, 1895; beside the books
of Weber and Schottky, for the case p = 3, already referred to (§§ 247, 199), and the book of
Krause for the case p = 2, referred to § 199, to which a bibliography is appended. References to
the literature of the theory of the transformation of theta functions are given in chapter XX.
In the papers of Schottky, in Crelle, en. and onwards, and the papers of Frobenius, in
Crelle, xcvn. and onwards, and in Humbert and Wirtinger (loc. cit. Ex. iv. p. 340), will be found
many results of interest, directed to much larger generalizations ; the reader may consult Weier-
strass, Berlin. Monatsber., Dec. 1869, and Crelle, LXXXIX. (1880), and subsequent chapters of the
present volume.
t References are given throughout, in footnotes, to the case where the characteristics are n-th
parts of integers. In these footnotes a capital letter, Q, denotes a characteristic whose elements
are of the form q'Jn, or of the form qjn, qt' , q^ being integers, which in the ' reduced ' case are
positive (or zero) and less than n. The abbreviations of the text are then immediately extended
to this case, n replacing 2.
295] DEFINITIONS OF SOME SYMBOLS. 487
whose elements si, st are given by S{ = q{ -I- r/, Si = qi + Ti. The charac
teristic, \t, wherein £/= s/, ti = S{ (mod. 2) and each of £/, ...,tp is either
0 or 1, is denoted by QR. Unless the contrary is stated it is intended in
any characteristic, $q, that each of the elements qj, qi is either 0 or 1. If
^q, ^r, ^k be any characteristics, we use the following abbreviations
p
Q\=-.qq' = q1q1'+ ...... +qpqp', i Q, R = qr' - q'r = 2 (g<r< - ft'r<),
t=i
|Q, R,K\ = \R, K + K, Q + Q, R , ( ®] = e^r = e^'r>+ - +M •
\-ft/
further we say that two characteristics are congruent when their elements
differ only by integers, and use for this relation the sign =. In this sense
the sum of two characteristics is congruent to their difference. And we
say that two characteristics Q, R are syzygetic or azygetic according as
| Q, .R = 0 or = 1 (mod. 2), and that three characteristics P, Q, R are
syzygetic or azygetic according as P, Q, R = 0 or = 1 (mod. 2).
Ex. Prove that the 2/> + l characteristics arising in § 202 associated with the half
periods ua' °\ ua' a>, ua' c% ..., u°" aP, ua> c are azygetic in pairs. Further that if any four of
these characteristics, A, B, C, D, be replaced by the four, BCD, CAD, ABD, ABC, the
statement remains true; and deduce that every two of the characteristics 1, 2, ..., 7 of
§ 205 are azygetic.
295. A preliminary lemma of which frequent application will be made
may be given at once. Let a,, ,, ..., a,, „, ..., ar> ,, ..., ar>n be integers, such
that the r linear forms
Ui = aitlxl + ...... + ais nxn, (i = 1, 2, . . . , r),
are linearly independent (mod. 2) for indeterminate values of xlt ...,#„;
then if Oi, ..., ar be arbitrary integers, the r congruences
f/i = «! , . . . , Ur = ar , (mod. 2),
have 2n~r sets of solutions* in which each of xlt ..., xn is either 0 or 1. For
consider the sum
2 [i + e^tf.-a.)] ... [1 + e^Wr- "'•>].
* *i, .-., xn
wherein the 2n terms are obtained by ascribing to xl, ..., xn every one of the
possible sets of values in which each of #,, ..., xn is either 0 or 1. A term in
which xl} ..., xn have a set of values which constitutes a solution of the
proposed congruences, has the value unity. A term in which xly ..., xn do
not constitute such a solution will vanish ; for one at least of its factors will
vanish. Hence the sum of this series gives the desired number of sets of
* When the forms C/j , . . . , Ur are linearly independent mod. m, the number of incongruent
2irr
sets of solutions is mn~r. In working with modulus m we use u = e m , instead of eiw ; and instead
of a factor i + e™(u<-«^ we U8e a factor 1 + M + ^ +- ... +/j.n~\ where n = t»Ul~n>.
488 PROOF OF A LEMMA. [295
solutions of the congruences. Now the general term of the series is typified
by such a term as
where /* may be 0, or 1, or . . . , or p ; and this is equal to
where
and, therefore, equal to
now, when //, > 0, one at least of the quantities clt ...,cn must be = 1 (mod. 2),
since otherwise the sum of the forms Ult ..., U^ is = 0 (mod. 2), contrary to
the hypothesis that the r forms Ult ..., Ur are independent (mod. 2); hence
the only terms of the summations which do not vanish are those arising for
fj, = 0, and the sum of the series is
Is i
Or >
* x
or 2n~r.
Ex. i. If, of all 22P half-integer characteristics, %q, the number of even characteristics
be denoted by g, and h be the number of odd characteristics, prove by the method here
followed that g-h, which is equal to Se™?9', is equal to 2". This equation, with g + h = 2*>,
determine the known numbers* g = ^-\ (2*> + l), h = <2P~l (2"- 1).
Ex. ii. If \a denote any half-integer characteristic other than zero, and %q become in
turn all the 2*> characteristics, the sum Se7™'1 ^' Ql = 2e™ ^-a'^ vanishes. For it is equal to
and if \a be other than zero, one at least of these factors vanishes. On the other hand it
is obvious that 2e™ ' °- Q ' = 22P.
We may deduce the result from the lemma of the text. For by what is there proved
there are 22*-1 characteristics for which \A, Q\ = 0 (mod. 2) and an equal number for
which | A, §| = 1.
296. We proceed now to obtain a group of characteristics which are
such that every two are syzygetic.
Let P! be any characteristic other than zero ; it can be taken in 22^ — 1
ways.
Let P2 be any characteristic other than zero and other than Pj , such that
P1}P2 =0(mod. 2);
* Among the n2? incongruent characteristics which are 7i-th parts of integers, there are
,,P-I (wp + r? - 1) for which Q \ = 0 (mod. n), and n?-1 (n? - 1) for which | Q = r (mod. n), when r
is not divisible by n.
296]
GOPEL GROUP OF CHARACTERISTICS.
489
by the previous lemma (§ 295), P2 can be taken in 2'*"1 — 2 ways ; also by
the definition, if P1P.1 be the reduced sum* of P,, P2,
P^PJP.H P^P, + P,, P,| = 0(mod. 2).
Let P3 be any characteristic, other than one of the four 0, PI, P2, PiP2,
such that the two congruences are satisfied
| P3, P! = 0, | P,, P2 = 0, (mod. 2) ;
then P3 can be chosen in 22*~2 — 22 ways ; also, by the definition,
| P3, P,P, | = P,, P, + P,, P2 = 0, (mod. 2),
| P,, P,P, | = 0, etc.
Let P4 be any characteristic, other than the 23 characteristics
0, Plf P2, P,, P.P,, P2P3, P.P,, P.P.P3,
which is such that
| P4, A = 0, P4, P2 = 0, P4, P3 = 0, (mod. 2) ;
and
then P4 can be chosen in 2^~3 - 2s ways, and we have
and
P2P3, P4 = P2, P4 + P3, P4 ! = 0, (mod. 2), etc.,
P,P,P,, P4 1 = Plf P4
, P4 = 0, (mod. 2).
P P I
*ti ft \ •
Proceeding thus we shall obtain a group of 2r characteristics,
0 P, Po P Po PPP
v> •*• 1) •*• 2 1 "••? •*• I* 2? • • • > -*• 1 -*• 2-* 3j • • * >
formed by the sums of r fundamental characteristics, and such that every
two are syzygetic. The r-th of the fundamental characteristics can be
chosen in 22?-r+1 - 2r~1 = 2r~l (2*p-*-+* - 1) ways; thus we may suppose r as
great as p, but not greater. Such a group will be denoted by a single
letter, (P) ; the r fundamental characteristics, P1} P2, P3, ..., may be called
the basis of the group. We have shewn that they can be chosen in
or
/ ff>*>Jl 1 \ /£V>»1 — O 1 \ / C*O*i 1 1 \ /flOl) ftf-L.it t \ Cl* I'/'f -i \ f I
I 1 I '"P -'I* __ I \ J^' \* — *•) j \f*
ways. But all these ways will not give a different group ; any r linearly
independent characteristics of the group may be regarded as forming a basis
of the group. For instance instead of the basis
we may take, as basis,
P P P
j. i, J. 2, • •• , ± r
PPP P
* I-1- 2> -*• 2) ••• t * r>
wherein PiP2 is taken instead of P, ; then Pl will arise by the combination
* So that the elements of PjPo are each either 0 or £.
490 GOPEL SYSTEMS [296
of PjP2 and P2. Hence, the number of ways in which, for a given group, a
basis of r characteristics, P/, ..., P/, may be selected is
(2*1 - 1) (2r - 2) . . . (2*- - 2'-1)/|r,
for the first of them, P/, may be chosen, other than 0, in 2r — 1 ways ; then
P2', other than 0 and P/, in 2r — 2 ways ; then P/ may be chosen, other than
0, P/, P/, P/Pa', in 2r — 22 ways, and so on, and the order in which they are
selected is immaterial.
Hence on the whole the number of different groups, of the form
0 p p p p P P P
Vj -L i) J- 21 •••) -L I-1- 2> •••> * I-4 I* 3> •••
of 2r characteristics, in which every two characteristics of the group are
syzygetic*, is
(2^ — 1) (22^-2 — 1) ...... (%ip-vr+2 — 1)
~ ~
Such a group may be called a Gopel group of 2r characteristics. The
name is often limited to the case when r=p, such groups having been
considered by Gopel for the case p = 2 (cf. § 221, Ex. i.).
297. We now form a set of 2r characteristics by adding an arbitrary
characteristic A to each of the characteristics of the group (P) just obtained ;
let P, Q, R be three characteristics of the group, and A', A", A'", the three
corresponding characteristics of the resulting set ; then
\A',A",A'" = \AP,AQ,AR\ = \P,Q,R = Q, R + R,P + P,Q , (mod. 2),
as is immediately verifiable from the definition of the symbols ; thus the
resulting set is such that every three of its characteristics are syzygetic, that
is, satisfy the condition
| A', A", A'" = 0, (mod. 2) ;
this set is not a group, in the sense so far employed ; we may choose r + 1
fundamental characteristics A, Al} ..., Ar, respectively equal to A, APl}
J.P2, ..., APr> and these will be said to constitute the basis of the system;
but the 2r characteristics of the system are formed from them by taking only
combinations which involve an odd number of the characteristics of the basis.
The characteristics of the basis are not necessarily independent ; there may,
for instance, exist the relation A + AP1 = AP2, or A ^P^. But there can
be no relation connecting an even number of the characteristics of the basis ;
for such a relation would involve a relation connecting the set, Pl} P2, . .., Pr,
of the group before considered, and such a relation was expressly excluded.
Hence it follows that there is at most one relation connecting an odd number
* When the characteristics are n-th parts of integers, the number of such syzygetic groups is
(n2"-!) ... (n2P-2r+2-l) divided by (nr-l) ... (n- 1).
297] OF CHARACTERISTICS. 491
of the characteristics of the basis ; for two such relations added together
would give a relation connecting an even number.
Conversely if A, Aly ..., Ar be any r+l characteristics, whereof no
even number are connected by a relation, such that every three of them
satisfy the relation
A', A", A'" = 0, (mod. 2),
we can, taking Pa = AaLA, obtain r independent characteristics P1( ..., Pr> of
which every two are syzygetic, and hence, can form such a group (P) of 2r
pairwise syzygetic characteristics as previously discussed. The aggregate of
the combinations of an odd number of the characteristics A, Alt ..., Ar may
be called a Gopel system* of characteristics. It is such that there exists no
relation connecting an even number of the characteristics which compose the
system, and every three of the 2r characteristics of the system satisfy the
conditions
| A', A", A'" = 0, (mod. 2).
We shall denote the Gopel system by (AP).
To pass from a definite group, (P), of 2r pairwise syzygetic characteristics
to a Gopel system, the characteristic A may be taken to be any one of the
2^ characteristics. But if it be taken to be any one of the characteristics of
the group (P), we shall obtain, for the Gopel system, only the group (P) ; and
more generally, if P denote in turn every one of the characteristics of the
group (P), and A be any assigned characteristic, each of the 2r characteristics
AP leads, from the group (P), to the same Gopel system. Hence, from a
given group (P) we obtain only 2'*~r Gopel systems. Hence the number of
Gopel systems is equal to
(2* -1) (£*" -1)... (2*-*+' -1)
We shall say that two characteristics, whose difference is a characteristic of
the group (P). are congruent, mod. (P). Thus there exist only 2P>-r
characteristics which are incongruent to one another, mod. (P).
It is to be noticed that the 22^~r Gopel systems derived from a given
group (P) have no characteristic in common; for if P1} P2 denote character
istics of the group, and A1} Az denote two values of the characteristic A, a
congruence A^ = A2P2 would give A^A^P^^, which is contrary to the
hypothesis that Al and A2 are incongruent, mod. (P). Thus the Gopel
systems derivable from a given group (P) constitute a division of the 2=*
possible characteristics into 2^~r systems, each of 2r characteristics. We can
however divide the 2^ characteristics into 2^-r systems based upon any
group (Q) of 2r characteristics ; it is not necessary that the characteristics of
the group (Q) be syzygetic in pairs.
By Frobenius, the name Gopel system is limited to the case when r = p.
492
GOPEL SYSTEMS CONSISTING WHOLLY
[297
Ex. For £> = 2, r = 2, the number of groups (P) given by the formula is 15. And the
number of Gopel systems derivable from each is 4. We have shewn in Example iv.,
§ 289, Chap. XV., how to form the 15 groups, and shewn how to form the systems
belonging to each one. The condition that two characteristics P, Q be syzygetic is equiva
lent to | PQ | = P | + 1 Q | (mod. 2), or in words, two characteristics are syzygetio when their
sum is even or odd according as they themselves are of the same or of different character.
It is immediately seen that the 15 groups given in § 289, Ex. iv., satisfy this condition.
The four systems derivable from any group were stated to consist of one system in which
all the characteristics are even and of three systems in which two are even and two odd.
We proceed to a generalization of this result.
298. Of the 22*'~r Gopel systems derivable from one group (P), there is a
certain definite number of systems consisting wholly of odd characteristics,
and a certain number consisting wholly of even characteristics*. We shall
prove in fact that when p>r there are 20""1 (2°" + 1) of the systems which
consist wholly of even characteristics, o- being p — r ; these may then be
described as even systems ; and there are 2<r~1 (2°" — 1) systems which may be
described as odd systems, consisting wholly of odd characteristics. When p = r,
there is one even system, and no odd system. In every one of the 22cr(2r — 1)
Gopel systems in which all the characteristics are not of the same character,
there are as many odd characteristics as even characteristics.
For, if Pj, ..., Pr be the basis of the group (P), a characteristic A which
is such that the characteristics A, APl} ..., APr are all either even or odd,
must satisfy the congruences
\XP,\= XP2 = = X , (mod. 2)
which are equivalent to
Y P- =\ P- a — i 9 T\
•<*-> -L I — \ -L I > \fc ~ 1> 6} •••>'}}
as is immediately obvious. Since, when X, Pl \ = j 7^ , and X, P2 1 = P
X, P,
X,P.2\+
= I Pj I + P2 + I PI, P2 I = !
etc., it follows that these r congruences are sufficient, as well as necessary.
These congruences have (§ 295) 22^~r solutions. If A be any solution, each
of the 2r characteristics forming the Gopel system (AP) is also a solution ;
for it follows immediately from the definition, if P, Q denote any two
characteristics of the group, that
\APQ\ = \A + P\+ Q + A, p +\A,Q + P, Q
= \A\ + 2\P +2JQ + P,Q\
= \A ,
because P, Q.\ = 0. Hence the 2'*~r solutions of the congruences consist of
* This result holds for characteristics which are -;t-th parts of integers, provided the group (P)
consist of characteristics in which either the upper line, or the lower line, of elements, are zeros.
299] OF ODD OR OF EVEN CHARACTERISTICS. 493
2w-'-f2r = <2?p-'»- characteristics A, and the characteristics derivable therefrom
by addition of the characteristics, other than 0, of the group (P) ; namely
they consist of the characteristics constituting 22^~2r Gopel systems, these
systems being all derived from the group (P). In a notation already
introduced, the congruences have 2^~2'' solutions which are incongruent
(mod. (P)).
Ex. If S be any characteristic which is syzygetic with every characteristic of the
group (P), without necessarily belonging to that group, prove that the 22P~2r characteristics
SA are incongruent (mod. P), and constitute a set precisely like the set formed by the
characteristics A.
299. Put now a = p — r, and consider, of the 22<r Gopel systems just
derived, each consisting wholly either of odd or of even characteristics,
how many there are which consist wholly of odd characteristics and how
many which consist wholly of even characteristics. Let h be the number of
odd systems, and g the number of even systems. Then we have, beside the
equation
g + h = 22-,
also
g — k = 2~-'"S,e'"'ilRl [ I+^^^.-P.I-^I^I] ... [i + e*i\it,pr\—*i\prn
wherein Plt ..., Pr are the basis of the group (P), and R is in turn every one
of the 22^ possible characteristics. For, noticing that the congruence
| RP = .ft | is the same as | R, P \ = \ P |, it is evident that the element of
the summation on the right-hand side has a zero factor when R is a
characteristic for which all of R, RPlt ..., RPr are not of the same
character, either even or odd, and that it is equal to 2~renilRl when
these characteristics are all of the same character; while, corresponding
to any value of R, say R = A, for which all of R, RP1} ..., .RPr are of
the same character, there arise, on the right hand, 2r values of R, the
elements of the Gopel set (AP), for which the same is true.
Now if we multiply out the right-hand side we obtain
wherein^ 2 denotes a summation extending to every set of /j, of the
/"i, PI, ...
characteristics P1( ..., PM, and /a is to have every value from 1 to ?•; but
we have, since P,, P.2, ... , are syzygetic in pairs,
and therefore
where S, = RP1 ... P^, will, as R becomes all 2^ characteristics in turn,
494 DETERMINATION OF GOPEL SYSTEMS [299
also become all characteristics in turn; also ^eniljl] = ^ewils^ is immediately
R s
seen to be 2^ ; it is in fact the difference between the whole number of even
and odd characteristics contained in the 22p characteristics. Hence
and therefore g-h = %>-r = 2".
This equation, with g + h = 22<r, when cr > 0, determines g = Z*~l (2°" + 1)
and h = 20""1 (2CT — 1), and when cr = 0 determines g = 1, h = 0.
These results will be compared with the numbers %>-1 (2^ + 1), 2p~l (2p - 1),
of the even and odd characteristics, which make up the 22p possible character
istics.
If Pi denote every characteristic of the group (P) in turn, and Pm denote
one characteristic of the bases P}, ..., Pr, and R be such a characteristic that
the 2r characteristics RPi are not all of the same character, at least one of
the r quantities R, Pm \ + \ Pm | is = 1 (mod. 2), and therefore the product
r
IJ H + 0iri\Pm\+iri\R,Pm\\
m=l
is zero. But, in virtue of the congruences,
I p.p. I — I P. I _l_ I P. I 17? P. I 4. 7? P. =17? P.P. I
I ***J I = I •* i \ ' I *} I ' I •"'» -*i I ' -**t *i = I •"') *** J I '
this product is equal to
2r 2r
^ girilPil +ni\R, Pi I Qr g-Tril^l ^ gTril/JPil _
Now e™ ' fip' ' is 1 or — 1 according as RPi is an even or odd characteristic.
Hence the system of 2r characteristics RPi contains as many odd as even
characteristics, and therefore 2r~l of each, unless all its characteristics be of
the same character.
300. The 22<r Gb'pel systems thus obtained, each of which consists wholly
of characteristics having the same character, either even or odd, have a
further analogy with the 22p single characteristics. We have shewn (§ 202,
Chap. XL) that the 2^ characteristics can all be formed as sums of not more
than p of 2p + 1 fundamental characteristics, whose sum is the zero character
istic; we proceed to shew that from the 22cr Gopel systems we can choose
2cr + 1 fundamental systems having a similar property for these 22<r systems.
Let the s = 22<r Gopel systems be represented by
(^P), ...,(2L.P),
the first of them, in a previous notation, consisting of Al and all characteristics
which are congruent to A± for the modulus (P), and similarly with the others.
Then we prove that it is possible, from A1} ..., As to choose 2<r + 1 character-
300]
WHEREOF EVERY THREE ARE AZYGETIC.
495
istics, which we may denote by Alt ..., A2^+1, such that every three of them,
say A', A", A"', satisfy the condition
j A', A", A'" | = 1, (mod. 2) ;
but it is necessary to notice that, if P be any characteristic of the group (P),
I A'P, A", A'" , = \A', A", A'" + | P, A" + P, A'"
is =
A',A",A'"\; for | P,
P , is also = P, A'" ; hence, if B', B", B'"
be any three characteristics chosen respectively from the systems (A'P),
(A"P), (A'"P), the condition | A', A", A'" \ = 1 will involve also Bf, B", B'" = I ;
hence we may state our theorem by saying that it is possible, from the
22<r Gopel systems, to choose 2<r -f 1 systems, whereof every three are azygetic.
Before proving the theorem it is convenient to prove a lemma ; if B be
any characteristic not contained in the group (P), in other words not
= 0 (mod. (P)), and R become in turn all the 22<r characteristics A1} ..., A8,
then*
•£e™ i R, B i _ o.
R
For let a characteristic be chosen to satisfy the r + 1 congruences
X, B =1, X, P! = 0, ...,|Z, Pr =0, (mod. 2),
and, corresponding to any characteristic R which is one of Alt ..., As, and
therefore satisfies the r congruences | R, P; | = P J , take a characteristic
S = RX; then
\8,B\- R,B = \X,B EEl.and S,Pt\= RX,Pi\ = \R,Pi\ + \X,Pi = P<|,
because | X, Pi = 0 ; hence the characteristics Alt ..., As can be divided into
pairs, such as R and S, which satisfy the equation eni ' s> B ' = — evi ' R> B '. This
provesf that ^e^R> B{ = 0.
R
We now prove the theorem enunciated. Let the characteristic Al be
chosen arbitrarily from the s characteristics Alt ..., As\ this is possible in
22<r ways. Let A2 be chosen, also from among A1} ...,AS, other than Al ;
this is possible in 22<r — 1 ways. Then A3 must be one of the characteristics
A-i, ...,AS> other than Alt A2, andj must satisfy the congruence | Al)Az, X \ =1.
The number of characteristics satisfying these conditions is equal to
* We have proved an analogous particular proposition, that if B be not the zero characteristic,
and R be in turn all the 2-'*> characteristics, Se™ ' R' B ' = 0 (§ 295, Ex. ii.).
R
t If R be all the 2*P characteristics in turn, Se1" ' 0> R ' = 2^. If P be one of the group (P),
and tf be one of Alt ... , A,, so that | R, P \ = \ P |, we have Se7"*'1 A * l = ewi ' P ' 22".
•
t We do not exclude the possibility ^as^^jj. Since l^,^,,, ^^,,1 = 1 Al, A.2 \, it is a
possibility only if |^lt A^\ = l.
496
DETERMINATION OF GOPEL SYSTEMS
[300
wherein R is in turn equal to all the characteristics Alt ..., As. For a term
of this series, in which R satisfies the conditions for A3, is equal to unity*,
while for other values of R the terms vanish. Now, since A1} A2, R
R, A1A2 \ + | A1} A2 , the series is equal to
the characteristic A1A2 cannot be one of the group (P), for if A1A2 = P, then
A2 = A1P> which is contrary to the hypothesis that Alt ..., As are incon-
gruent for the modulus (P); hence by the lemma just proved the sum of the
series is 2'2<r~1, and A3 can be chosen in 220""1 ways.
We consider next in how many ways A4 can be chosen ; it must be one of
A1} ..., As other than Alt A2, A3 and must satisfy the congruences
i A1} A2, X I = 1, | Ait A3, X = 1,
which, in virtue of the congruence A1} A2, A3 = 1, and the identity
A2, A3) X
, Alt X
, X
ly A2, A3
involve also j A2, As, X = 1. The number of characteristics which satisfy
these conditions is equal to
or
2, R\_ -2
where R is in turn equal to every one of A1} ..., As; hence, in virtue of the
lemma proved, using the equations,
A1}
2, R
AltA2
R,
j, A3 + A2AS, R
the number of solutions obtained is 22a~2. But we have
\A1A2A3,A1,A2
A1A.2A3,A1A
so that AiA^Az also satisfies the conditions.
Now it is to be noticed that, for an odd number of characteristics
Blt ..., -B^+1, the condition that every three be azygetic excludes the
possibility of the existence of any relation connecting an even number of
these characteristics, and for an even number of characteristics B1} ...,5^,
the condition that every three be azygetic excludes the possibility of the
existence of any relation connecting an even number except the relation
B1B2 . . . B^ = 0. For, B being any one of Blt ..., -5o*+i other than Blt ..., B2m,
we have, as is easy to verify,
B1B2 . . .
•\BltBtm,B\-^B9tBm,tB
It is immediately seen that A, B, B \ = 0.
, B
300]
WHEREOF EVERY THREE ARE AZYGETIC.
497
so that the left hand is = 1 ; therefore, as B2m, B2m, B = 0, we cannot have
B2m = B1B2 ... #>,„_!• This holds for all values of m not greater than k, and
proves the statement.
Hence, 2o-+ 1 being greater than 4, we cannot have A4 = A1A2A3, for we
are determining an odd number, 2cr + l, of characteristics. On the whole,
then, A 4 can be chosen in 22<7~2 — 1 ways.
To find the number of ways in which A5 can be chosen we consider the
congruences
I, A2, .A. =1, AI, A3, JL = 1,
, 2L \ = 1,
which include such congruences as A2, A3, X \ = 1, A2, A±, X\ = 1, etc.
The characteristic A5 must be one of Alt ..., As, other than Al} A2, A3, At\
the condition that A5 be not the sum of any three of AI, A2> A3, A4 is
included in these conditions. The number of ways in which As can be
chosen is therefore
where R is in turn equal to every one of Alt ..., As; making use of the fact
that A1A2A3A4 is not = 0, we find the number of ways to be 22<r~3.
Proceeding in this way, we find that a characteristic A2m+1 can be chosen
in a number of ways equal to the sum of a series of the form
2~ (zrn-i) 2 ("l — e1"!-4!' ^2' ^H [1 — e1™!-4!' •4»«-Ri"| ... fl — e1"!-4!' Azm> R\~\
R
and therefore in 22(7~<2m~1) ways, and that a characteristic A^ can be chosen
in 22<r-<2m-2) — 1 ways, the value Azm = A1 A2 ... A2m^ being excluded. In
particular A^ can be chosen in 22 — 1 ways, and -42<r+i in 2 ways.
To the 2<r+l characteristics thus determined it is convenient* to add
the characteristic A2<r+2 = A1A2 . . . A2(r+1 ; if Ai} Aj be any two of Al} ..., A2ff+l
we have
A2<r+2t A{, Aj = I A{, Aj, AI | + ...... + | A{, Aj, A2a-+i \ = 1,
the expressions Ai} Aj, At \, \ Ait Aj, Aj \ being both zero. We have then
the result : From the 2'2<r characteristics Al} ..., As it is possible to choose a
set AI, ..., -42<H-2> such thut every three of them satisfy the condition
\A',A",A'" =1,
in
1 (22ff~2 - 1) ... (22- 1) 2 2*r+ff8 (2ar- 1) (22ff~2 - 1) ... (22- 1)
ways ; there exists no relation connecting an even number of the characteristics
Al} ..., -A2<r+2 except the prescribed condition that their sum is zei*o ; since the
sum of two relations each connecting an odd number is a relation connecting
* In the particular case of § 202, Chap. XI., A2tT+2 is zero.
B. 32
498 ROOT SETS. [300
an even number, there can be at most* only one independent relation con
necting an odd number of the characteristics A!, ..., -42<7+2. And, as before
remarked, to every one of the characteristics Al} ..., -42<r+2 is associated a
Gopel system of 2r characteristics.
301. The 22or systems. (A1P), ..., (ASP), which have been considered,
were obtained by limiting our attention to one group (P) of 2r pairvvise
syzygetic characteristics. We are now to limit our attention still further to
the sets Alt ..., -A2<r+o just obtained satisfying the condition that every three
are azygetic.
If to any one of the characteristics Al} ..., A2<T+2, say A^, we add the
characteristic X, the conditions that the resulting characteristic may still
be a characteristic of the set Alt ..., As, are (§ 298) the r congruences
XAk, Pi | = j Pi \, in which i= 1, ..., r ; in virtue of the conditions | A^, Pi \
= | Pi , these are equivalent to the r congruences \X, Pi \ = 0, which are
independent of k; these latter congruences have 2^~r solutions, but from
any solution we can obtain 2r others by adding to it all the characteristics of
the group (P). There are therefore 22p~2r = 22<T congruences X, incongruent
with respect to the modulus (P), each of which -f, added to the set Alf . . . , A2(r+z,
will give rise to a set AS, ..., A'2(T+2, also belonging to Alt ..., As. Further
| Ai, A/, Ajc' \ = XAi, XAj, XAk \ = Ai} Aj, Ak = 1 ; and any relation con
necting an even number of the characteristics J./> ..., A'2lT+2 gives a relation
connecting the corresponding characteristics of Alf ..., A^+2. Thus the
22<r sets derivable from A1} ..., A^+2 have the same properties as the set
A A
XI i, ... , ,TL2o.-|_2.
Hence all the sets Alt ..., AZtT+2 can be derived from
root sets by adding any one of the 22<r characteristics X to each characteristic
of the root set.
302. Fixing attention upon one of these root sets, and selecting
arbitrarily 2<7 + 1 of its characteristics, which shall be those denoted by
Aly ..., A2(r+i, we proceed to shew that of the 22<r characteristics X, there is
just one such that the characteristics XA1} ..., XA2<T+l, derived from
Alf ..., A2tr+1, have all the same character, either even or odd. The
conditions for this are
| .A.n.1 = XAf
* If the characteristic of which all the elements, except the z'-th element of the first line, are
zero, be denoted by £/, and E^ denote the characteristic in which all the elements are zero
except the t-th element of the second line, every possible characteristic is clearly a linear aggre
gate of Ejf, ... , Ep, El, ..., Ep. Thus when a has its greatest value, =p, there is certainly one
relation, at least, connecting any 2o- + 1 characteristics.
t It is only in case all the characteristics of the group (P) are even that the values of A" can
be the characteristics Al , ... , Ag.
302] FUNDAMENTAL SETS OF THE SAME CHARACTER. 499
which are equivalent to the 20- congruences
X.A.Ai = A^ + lAil (i=2,3, ...,(2<r + l));
if X be a solution of these congruences, and P be any characteristic of the
group (P), we have
XP,A,Ai\ = X,AlAi\ + \P,Al +\P,At = A, + Ai\ + 2\P\,
so that XP is also a solution; since the other congruences satisfied by X
were in number r, and similarly, associated with any solution, there were 2r
other solutions congruent to one another in regard to the group (P), it
follows that the total number of .characteristics X satisfying all the
conditions is %#-r-'**-r = l. Thus, as stated, from any 2<7+ 1 characteristics,
AI, ..., -£0.7+1, of a root set, we can derive one set of 2<r + l characteristics
Aly ...^A2<r+1, which are all of the same character, their values being of the
form Ai = XAi.
Starting from the same root set, and selecting, in place of Alt ..., Av+lt
another set of 2<r+l characteristics, say A^, ..., Av+z, we can similarly
derive a set of the form
X A2, ..., X 4lsrffi
consisting of 2cr + 1 characteristics of the same character. The question
arises whether this can be the same set as Alt ..., A.M+1. The answer is in
the negative. For if the set X'Aa, ..., X'A2<r+2 be in some order the same as
the set XAlt ..., XA2<r+l, or the set XX' A2, ..., XX'A2<r+2 the same as the
set Alt ..., A^+1, it follows by addition that XX' A^ = A^+2 or XX' = A^A*^.
Thence the set A.A.A^,, A.A-.A,^, ..., A^A^A^, Al is the same as
Alt A.,, ..., A^+l, or we have 2<7 equations of the form A^A^^^Aj, in
which i = 2, ..., 2a+l,j = 2, ..., 2<r + l. Since there is no relation con
necting an even number of the characteristics Alt ..., A2<T+2 except the one
expressing that their sum is 0, these equations are impossible*.
Similarly the question may arise whether such a set as Alt ..., A»+l, of
2o- + l characteristics of the same character, azygetic in threes, subject to no
relation connecting an even number, and incongruent for modulus (P), can
arise from two different root sets. The answer is again in the negative.
For if A1} ..., A.&+1, and B1} ..., Bw+l be two sets taken from different root
sets, the 2«r+l conditions XAi = X'Bi, for i = l, ..., 2<r+l, to which by
addition may be added XA^^X'B^^, shew that the set Blt ..., 52<r+2 is
derivable from the set A ,,..., A2IT+2 by addition of the characteristic XX' to
every constituent. This is contrary to the definition of root sets. Conversely
if AI, ..., A'2a+» be any one of the 2^ sets which are derivable from the root
set Alt ..., Av+a by equations of the form A-=ZAi, the set of 2<r + l
To the sets Alt ..., A^^ and X'A2, ..., X'A2<r+2 we may adjoin respectively their respective
sums. The two sets of 2ff + 2 characteristics thus obtained are not necessarily the same. When
ff is odd they cannot be the same, as will appear below (§ 303).
32—2
500 UTILITY OF FUNDAMENTAL SETS. [302
characteristics of the same character, say AI, . .., A'2<r+l, which are derivable
from AI, ..., -A'ar+i by equations of the form Ai = X'Al, will also be derived
from A1} ..., A2a+l by the equations AJ = XAi} in which X = X'Z.
On the whole then it follows that there are
different sets, Aly ...,A2<T+1, of 2<r + l characteristics of the same character,
azygetic in threes, subject to no relation connecting an even number, and
incongruent for the modulus (P).
Of the characteristics A1} ..., A2<T+1 there can be formed
(2<r + 1, 1) + (2<r+ 1, 3) + ... + (2cr + 1, 2(7 + 1) = 22"
combinations*, each consisting of an odd number ; and, since there is no
relation connecting an even number of Al} ..., Aw+l, no two of these com
binations can be equal. These combinations all belong to the characteristics
Alt ..., As, satisfying the r congruences X, Pf = | Pt \ ; for
I ~A ~A ~A PI— T P I _i_ _i_ I /T PI — IP
| -0.1-0.2 ••• •"•2fc— 1> •* i \ — •"•!> ri I T ••• T I •"•<&— 1> * < I = I J t •
And no two of them are congruent in regard to the modulus (P) ; for a
relation of the form
AI . • • -a.2k— i — •^•m-^-m+i • • • •"•m+2jA-*>
wherein P is a characteristic of the group (P), would lead to a relation of the
form A2p = A1A2...A2p_1P, and thence give
whereas
P A A
p—l-L , -0.2p, -tlj
A1 ... A2p_ly A.2p, A
.2p, 2p+1
T T ~A 7 i
-"•i • •• -"2p-i> •/12p> -"ap+i |
-"•!) -"-2p> -"2p+l H~ • • • + -"-2p— 1 » -"-2p,
Thus the 22(r combinations, each consisting of an odd number of the
characteristics A1} ..., -4ar+i> are in fact the characteristics ^4a, ..., As. We*f*
call the set A1} ...,Aw+l & fundamental set. We may associate therewith
the characteristic -42<r+2 = .41 ... A2a+l, which is azygetic with every two of the
set A1} ..., Aw+l; the case in which it has the same character as these will
appear in the next article. And it should be remarked that the argument
establishes, for the 22<r Gopel systems (A1P), ..., (ASP), the existence of
fundamental sets, (yljP), ..., (A2(r+lP), which are Gopel systems, by the odd
combinations of the constituents of which, the constituents of the systems
(A-iP), ..., (ASP) can be represented.
* Where (n, k) denotes n (n- l)...(?i- k+ l)/k I
t By Frobenius the term Fundamental Set is applied to any 2o- + 2 characteristics (incon
gruent mod. (P)) of which every three are azygetic.
303J THEIR CHARACTER. 501
303. The characteristics Aly ..., A.^+i have been derived to have the
same character. We proceed to shew now, in conclusion, that this character
is the same for every one of the possible fundamental sets, and depends only
on <r. Let (-7) be the usual sign which is +1 or —1 according as a- is a
\4/
quadratic residue of 4 or not, in other words, (-) = 1 when a- is = 1 or
\4/
= 0 (mod. 4), and ( y j = — 1 when & is = 2 or = 3 (mod. 4) ; then the character
of the sets Alt ..., Aw+l is |-J , that is, A1} ...,Aw+l are even when f-J = + 1
and are otherwise odd, and the character of the sum A2<r+2 = Al ... Ay,+\ is
e™ I -7 ) • Or, we may say
when er = 1 (mod. 4), Alt . .., A^+1 are even, A.2a+.2 is odd ;
when a = 0, Alt ..., A2a+l are even, -4^+2 is even,
when a = 2 (mod. 4), Alf ..., AZ<7+1 are odd, A2>,+<, is odd ;
when a = 3, AI} ..., Aw+l are odd, -4^+2 is even.
For if Aly ..., A2<T+l be all of character e we have
| Al A2 ... A*+1 1 = | lj | + ... + | Z*+1 1 + 2 | Ait AJ |,
where ^1^, AJ consist of every pair from Aly ..., A^^ ; also
where ^Ij, 4;, -4/, consist of every triad from Alt ..., -4afc+il hence, since
\Ai, AJ, Ah\ = l, and, as is easily seen, n(n — \}(n— 2)/3 ! is even or odd
according as n is of the form 4m + 1 or 4m + 3, it follows that S | -4{, -4j | is
even or odd according as 2& + 1 is of the form 4>m + 1 or 4m + 3 ; therefore
A1AZ ... A&+1 has the character e or — e according, as 2k + 1=1 or
= 3 (mod. 4). Thus the number of combinations of an odd number from
AI, ..., -4ar+i which have the character e is
+ 1,5) + (2o-+ 1,9) + ...
= i {(1 + «)2<r+1 - (1 - «)2<r+1 + 1 (1 - tV)2<r+1 - i (1 + ^)2<7+1}a;=1
= 22'-1 + 2*-* sin TT ;
4
this number is 220""1 + 2"7"1 when <r = 0 or a = 1 (mod. 4) ; otherwise it is
2'20""1 — 2"7"1 ; now we have shewn (§ 298) that the characteristics A1} ..., ^.g
contain respectively 22"7"1 + 20""1, 2'2"7"1 — 20""1 even and odd characteristics, and
(§ 302) that every one of Alt ..., As can be formed as an odd combina
tion from Alt ...,Ayr+l; hence e = + l when a = 0 or o- = l (mod. 4), and
502 SUMMARY OF RESULTS. [303
otherwise e=— 1; this agrees with the statement made. Further, by the same
argument A^AZ . . . Aw+l has the character e or — e according as 2o- + 1 = 1
or = 3 (mod. 4) ; and this leads to the statement made for A2^+2.
The reader will find it convenient to remember that the combinations,
from the fundamental set Al} ..., A.^+1) consisting of 1, 5, 9, 13, ... of them,
are all of the same character, and the combinations consisting of 3, 7, 11, ...
are all of the opposite character.
Ex. If Alt ..., ^42p + i be half-integer characteristics azygetic in pairs, and S be the
sum of the odd ones of these, prove that a characteristic formed by adding S to a sum of
any p + r characteristics of these is even when r=0 or =1 (mod. 4), and odd when r=2 or
= 3 (mod. 4). (Stahl, Crelle, LXXXVIII. (1879), p. 273.)
304. It is desirable now to frame a connected statement of the results
thus obtained. It is possible, in
(2*? - 1) (22^-2 - 1) ... (2-^-2'-+2 - l)/(2»> - 1) (2'-1 - 1) ... (2 - 1)
ways, to form a group,
0 p p p p P P P
UJ •*- 1» -1- 2> •••> *!•* iJ •••» •*,!•* t-*»l •••
of 2r characteristics, consisting of the combinations of r independent charac
teristics Pa, ..., Pr, such that every two characteristics P, P' of the group
are syzygetic, that is, satisfy the congruence | P, P' | = 0, (mod. 2). Such a
group is denoted by (P), and two characteristics whose difference is a
characteristic of the group are said to be congruent for the modulus (P).
From such a group (P), by adding the same characteristic A to each
constituent, we form a system, which we call a Gb'pel system, consisting of
the combinations of an odd number of r+ 1 characteristics A, AP1} ..., APr,
among an even number of which there exists no relation ; this system is such
that every three of its constituents, say L, M, N, satisfy the congruence
L, M, N \ = 0, or, as we say, are syzygetic. Such a Gb'pel system is
represented by (AP).
It is shewn that by taking 22^~r different values of A and retaining the
same group (P), we can thus divide the 22^ possible characteristics into
<pp-r Gopel systems. Among these %&-r Gopel systems there are 22P~2r
systems of which all the elements have the same character. Putting
2p — 2r = 2<r we shew further that 2°"~1 (2a + 1) of these Gopel systems
consist wholly of even characteristics, and that 2<r~1(2cr — 1) of them consist
wholly of odd characteristics. Putting s = 22<r we denote the 22<r Gopel
systems which have a distinct character by (^iP), ..., (ASP); and, still
retaining the same group (P), we proceed to consider how to represent these
22<r systems by means of 2cr -f 1 fundamental systems.
It appears then that from the characteristics A1} ..., As we can choose
2o--l-l characteristics Alf ..., A2<T+1 in
2 - 1) ... (22 -
305] EXAMPLES. 503
ways, such that every three of them are azygetic, and all have the same
character; this character is not at our disposal but is that of (- J ; the sum
\ T? /
of Alt ..., A-a+i, denoted by A2<r+2, has the character e™^). Then all the
\4v
combinations of 1, 5, 9, ... of A1} ..., Aw^ have the character f^j. and all
\4/
the combinations of 3, 7, 11, ... have the opposite character. These combi
nations in^ their aggregate are the characteristics Alt ..., As. The charac
teristics Alt ..., A2(r+1 are, like A1} ..., At, incongruent for the modulus (P).
To each of them, say Aiy corresponds a Gopel system (AiP), to any con
stituent of which statements may be applied analogous to those made for Zf
itself.
The characteristic A^+.2 is such that every three of the set Alt ..., A^+.2
are azygetic. This set is in fact derived, as one of 2cr + 2 such, from a set of
2<r + 2 characteristics, here called a root set, which satisfies the condition
that every three of its constituents are azygetic without satisfying the
condition that 2<r + 1 of them are of the same character. There are
such root sets. It is not possible, from any root set, to obtain another by
adding the same characteristic to each constituent of the former set.
The root sets are not the most general possible sets of 2<r + 2 charac
teristics of which every three are azygetic. Of such sets there are
2«r«+,'a (2*r - 1) . . . (22 - I)/ 2CT + 2,
but they break up into batches of 22<r, each derivable from a root set by the
addition of a proper characteristic to all the constituents of the root set.
305. As examples of the foregoing theory we consider now the cases <r = 0, <r= 1, o- = 2,
<r=p. When <r = 0, the number of Gopel groups of 2" pairwise syzygetic characteristics is
(2"+l)(2J>-i + l) ...... (2 + 1);
from any such group we can, by the addition of the same characteristic to each of its
constituents obtain one Gopel system consisting wholly of characteristics of the same even
character. These results have already been obtained in case p = 2 (§ 289, Ex. iv.),
and, as in that particular case, the 2f>- 1 other systems obtainable from the Gopel group
by the addition of the same characteristic to each constituent, contain as many odd
characteristics as even characteristics.
When o-=l, we can, from any Gopel group of 2"-1 pairwise syzygetic characteristics,
obtain 4 Gopel systems, three of them consisting of 2"-i even characteristics and one of
2* -i odd characteristics. The characteristics of the latter (odd) system are obtainable as
the sums of three characteristics taken one from each of the three even systems.
Whon o- = 2, the number of fundamental sets Alt ..., A& is
504 THE CASE WHEN a IS 3. [305
each of them has the character (jj, or is odd, and their sum, A6, is odd. Among the
22*=16 characteristics A1, ..., As there are ^2cr~1-2<r~1 or 6 odd characteristics; these
clearly consist of the characteristics Alf ..., A6 ; the six fundamental sets are obtained by
neglecting each of Alt ..., A6 in turn. Among the characteristics Alt ..., As there are 10
even characteristics, obtainable by combining A^ ..., Ag in threes. And, to each of the
characteristics Alt ..., Ag corresponds a Gopel system of 2r=2P-<r = 2^'~2 characteristics,
for the constituents of which similar statements may be made.
Of the cases for which a- = 2, the case p = 2 is the simplest. After what has been said
in Chap. XL, and elsewhere, we can leave that case aside here. For jo = 3 the Gopel
systems consist of two characteristics ; adopting, for instance, as the group (P), the pair
^ \OOOJ ' ^ (loo) ' ^G conc^tion for the characteristics Al, ..., A^ namely | X, Pl = P1\,
reduces to the condition that the first element of the upper row of the characteristic
symbol of X shall be zero ; hence the 16 characteristics Alf ..., A, may be taken to be
i (n 1 2 ) » wnere if1 2 ) represents in turn all the characteristic symbols for p = 2.
\0 QI a2 / • \QI a2 /
Taking next the case o- = 3, there are s=22<r=64 Gopel systems, (AP), each consisting
wholly either of odd characteristics or of even characteristics, there being 2"-1 (2* - 1), = 28,
odd systems, and 36 even systems. From the representatives, Alt ..., Ag, of these systems,
which are incongruent mod. (P), we can choose a fundamental set of 7 characteristics
Alt ...,Af in
29 (26-1) (2*- 1) (22-1) _
17 » —zoo,
ways; Alt ..., Aj will be odd, and their sum, J8, will be even; for (?J = (£)=-!,
6™ \4J = 1' ^e set ^n '"' ^7> ^8 ^s> ^n accor(iance witn ^e theory, derived from one
of 288/(2o- + 2), =36, root sets A»_...,A% (§ 301), by equations of the form A^XAi, in
which X is so chosen that A1} ..., A7 are of the same character ; from this root set we can
similarly derive 8 fundamental sets of seven odd characteristics, according as it is A6 or is
one of Alt ..., A7 which is left aside. Now the fact is, that, in whichever of the eight
ways we pass from the root set to the seven fundamental odd characteristics, the sum of
these seven fundamental characteristics is the same. We see this immediately in an
indirect way. Let A lt ..., Aj be a fundamental set of odd characteristics derived from the
root_set Alt ...,A8_by_ the equations A^XAi; putting A8 = A1... J7, consider the set
As, A^A^A^ ..., A^AVA^ Alt derived from A1, ..., J8 by adding A^A^ to each ; in the first
place it consists of one even characteristic, J8, and seven odd characteristics ; for
i, At\= At,Ai,At\ = l, (mod. 2),
because Alt ..., A& are azygetic in threes ; in the next place
I AS) A1} A^AiAi\m\ As, AH ^| = 1,
so that^every three of its constituents are azygetic. Hence the characteristics As A1A2,
..., A^Aj^Aj, Aj, which, as easy to see, are not congruent to Alt ..., Aj mod. (P), form,
equally with Aly ..., A7, a fundamental set, whose sum is likewise As ; they are derived
from Alt ..., A8 by adding A^A-^X to each of these. There are clearly six other jsuch
fundamental sets, derived from Aly ..., As by adding respectively ASA2X, ..., A8A^X.
Hence to each of the 36 root sets there corresponds a certain even characteristic and to
each of these even characteristics there correspond 8 fundamental sets. We can now shew
further that the even characteristics, thus associated each with one of the 36 root sets, are
306] APPLICATION TO THETA FUNCTIONS. 505
in fact the 36 possible* even characteristics of the set Jj, ..., As. This again we shew
indirectly by shewing how to form the remaining 7 . 36 fundamental systems from the
system Alt ..., An. The seven characteristics AKAZA3, ASA3A1, A^Aj^A^, At, A^, A6, A7,
are in fact incongruent mod. (P), they are all odd, have for sum A1A2A3, which is even,
and are azygetic in threes ; for ASA2A3 is a combination of five of A l , . . . , A 7 , and
14, Z5|+|13, I4, 16 =1, | Z4, J6, I6| = l,
(the modulus in each case being 2) ; hence these seven characteristics form a fundamental
system. There are 35 sets of three characteristics, such as Alt A2, A3, derivable from the
seven Alt ..., A7 ; each of these corresponds to such a fundamental system as that just
explained ; and each of these fundamental systems is associated with seven other funda
mental systems, derived from it by the process whereby the set A{, Z<ZS14S, ..., AiAsA7
is derived from A^ ..., A7.
When or =/>, a Gopel system consists of one characteristic only ; we can, in
/p\
ways, determine a set of 2/> + l characteristics, all of character ( ^ J , of which every three
/%)\
are azygetic ; their sum will be of character e"ip I ^ ) > a^ ^he possible 2%' characteristics
can be represented as combinations of an odd number of these.
306. We pass now to some applications of the foregoing theory to the
theta functions. The results obtained are based upon the consideration of the
theta function of the second order defined by
</> (M, a ; %q) = ^ (u + a ; %q) ^ (u - a ; %q),
where ^q is a half-integer characteristic; as theta function of the second
order this function has zero characteristic ; the addition of any integers to
the elements of the characteristic ^q does not affect the value of the function.
By means of the formulae (§ 190, Chap. X.),
wherein N denotes a row of integers and X(?*; s) = Hii(u + ^fls) — iriss, we
immediately find
> a ; 1 q) = ^ **> <f> (u, a ;
where %kq denotes the sum of the characteristics %k, ^q; to save the repeti
tion of the ^, this equation will in future be written in the form (cf. § 294)
0 (u
when the contrary is not stated capital letters will denote half-integer
characteristics, and KQ will denote the reduced sum of the characteristics
K, Q, having for each of its elements either 0 or £.
* Thus, when p = 3=ff, the result quoted in § 205, Chap. XI., is justified.
506 A THETA FUNCTION OF THE SECOND ORDER [306
We shall be concerned with groups of 2'' pairwise syzygetic characteristics,
such as have been called Gopel groups, and denoted by (P) ; corresponding
to the r characteristics P1( ...,Pr from which such a group is formed, we
introduce r fourth roots of unity, denoted by en ..., er, which are such that
6 2 _ Qiri I P, I ? _ ^2 _ ewi I Pf I .
the signs of these symbols are, at starting, arbitrary, but are to be the same
throughout unless the contrary be stated. Since the characteristics of the
group (P) satisfy the conditions
| Pi, Pj \ = 0, (mod. 2),
we may, without ambiguity, associate with the compound characteristics of
the group the 2r — r symbols denned by
eo=l, ei)j = ei
i Pi \ /PA /PA /PA / P,- \ / P* \
*, ;, * = *<?;, k (p p J = ef 6; 6A I I 1 (I = 6J6*. ; [ppj = €* €f, j ( p p I ,
\fjfJt/ \-L k/ V* »/ V-1- j' x-4 ik-^ t' v* t-1 7'
.
and ej = et-f ^ «= e^e^ I p , etc.
V *<
Consider now the function* denned by
where A is an arbitrary half-integer characteristic, and Pi denotes in turn all
the 2r characteristics of the group (P). Adding to u a half-period Qpk,
corresponding to a characteristic Pk of the group (P), we obtain
<t> (u 4- nPjfc, a; A) —
if then P^ = PiP*, or P{ = PhPk, we have
P\ / P, \ /Pi\ /P, \ /P, \ /P,\ /P,\ /P..
i\ I * 1 \ I •rh\ I J- k\ I * k\ (fm\ f •* il I r h , . „
— fcfcc
now, as Pi becomes in turn all the characteristics of the group (P), Ph, = PiPk,
also becomes all the characteristics of the group, in general in a different
order ; thus we have
>,«; A),
* If preferred the sign ( ) , whose value is ± 1, may be absorbed in ej . But there is a
tain convenience in writing it explicitly.
307] WITH A GOI'EL SYSTEM AS PERIODS. 507
If 2fl3[ be any period, we immediately find
4> (w + 2&M, a I -4) = e2A(W; m <& (u, a ; A).
Thus, X(w; P*) being a linear function of the arguments %, ..., up, the
function <3> (w, a; -4) is a theta function of the second order with zero
characteristic, having the additional property that all the partial differential
coefficients of its logarithm, of the second order, have the 2r sets of simul
taneous periods denoted by the symbols ^ipk-
Ex. i. If S be a half-integer characteristic which in syzygetic with every characteristic
of the group (/*), prove that
* (u + Q,, a ; A) = e*(u<
and
jfilt1. ii. If Pj; be any characteristic of the group (1*), prove that
4> (u, a • Al\} = ( «-i * («, a ; ^1).
j&a;. iii. When, as in Ex. i., S is syzygetic with every characteristic of the group (P),
shew that
e-'IS'V * («, a ; APk) * (•», Z> ; APt) = e*i{s* *(M, a ; J) * (», 6 ; ^1).
Conversely it can be shewn that if a theta function of the second order
with zero characteristic, H (w), which, therefore, satisfies the equation
II(tt+fl,») = e2A'»<M) n(«),
for integral m, be further such that for each of the two half-periods associated
with the characteristics \m = P, ^m = Q, there exists an equation of the form
n (u + %tim) = e^+"1«i+-+"/.«;. n (u),
where p, vl} ..., vp are independent of u, then the characteristics P, Q must
be syzygetic. Putting vii = v1n1+ ...... + vpup, we infer from the equation
just written that
n (u + nm) = ^+-(«+i«m) n (u + ^nw) = &+*™+wm n (M) ;
comparing this with the equation
n (u + n,a) = e2A'»(M) n (M) = e*u™<H+*a>»>-*rimm' n (%)
we infer that v = Hm, p = kiri + ^HmQm — 'iriiniri, where k is integral, and
hence
n (u + |nm) = + e-i-*»M»'+2A<us J?w) n ^w^
307. In accordance with these indications, let Q(u) denote an analytical
integral function of the arguments w1} ..., up which satisfies the equations
Q(u + nm) = ^"'^' Q(u); Q(w + ftPjfc) = efce«i^i+^«5^> Q(M),
for every integral m and every half-integer characteristic P^ of the group (P).
508 GENERAL EXPRESSION OF SUCH A FUNCTION. [307
We may regard the group (P) as consisting of part of a group of 2^
painvise syzygetic characteristics formed by all the combinations of the
constituents of the group (P) with the constituents of another pairwise
syzygetic group (R) of %P~r characteristics. Then the 2? characteristics of
the compound group are obtainable in the form PiRj, wherein P{ has the 2r
values of the group (P), and Rj has the 2^~r values of the group (R). Since
every 2^+1 theta functions of the second order and the same characteristic
are connected by a linear equation, we have
CQ(u)=2Ci>j<f>(u,a;
where C, Gitj are independent of u and are not all zero*. Hence, adding to
u the half-period Qpk, we have
*j Q (u) = 2 Citj e^-> ^ ^ LJ 4> (u, a ; P{PkRj),
and therefore, as q,^*'p*J = e"1,
/ p \
CQ (u) = 2 Cij (p »- J e*</> (w, a ; PiPkRj) ;
forming this equation for each of the 2r values of Pk, and adding the results,
we have
herein put Ph = PiPk, so that as, for any value of i, Pk becomes in turn all
the characteristics of the group (P), the characteristic P^ also becomes all the
characteristics in turn, in general in a different order ; then
Ph\ /PhPi\ /Ph\ /P
m = €h€i
and, therefore,
u) = 2 2 eh
j h
where
*
and thus
Now the 2^~r functions <E> (tt, a ; Rj) are not in general connected by any
linear relation with coefficients independent of u ; for such a relation would
be of the form
* It is proved below (§ 308) that the functions <f> (u, a ; P{Rj) are linearly independent, so
that, in fact, C is not zero.
308] THE GENERAL THEOREM. 509
wherein Hi is independent of u, and Qi becomes, in turn, all the constituents
of a group (Q) of 2^ pairwise syzygetic characteristics, and we shall prove (in
§ 308) that such a relation is impossible for general values of the arguments
a. Hence, all theta functions of the second order, with, zero characteristic,
which satisfy the equation
Q(u + flPjfc) = ete«'ip*i+2A<«s *V Q (u)
for every half-integer characteristic Pk of the group (P), are representable
linearly by 2^"*", = 2", of them, with coefficients independent of u. We have
shewn that the functions <I> (u, a; A), defined by the equation
u-a; APt),
where the summation includes 2r terms, are a particular case of such theta
functions.
308. Suppose there exists a relation of the form
where the summation extends to all the 2?) characteristics Q{ of a Gopel group (Q), and Hi
is independent of u. Putting for u, it + QQa, where Qa is a characteristic of the group (Q),
we obtain
hence, if tlt ..., ep are fourth roots of unity associated with a basis Qlt ..., Qp of the group
(Q), as before, and this equation be multiplied by fa, and the equations of this form
obtained by taking Qa to be, in turn, all the 2p characteristics of the group (Q), be added
together, we have
now let Qj-=QaQi} then for any value of i, as Qa becomes all the characteristics of the
group (Q), Qj will become all those characteristics ; therefore, substituting
we have
hence one at least of the expressions
2f>3 (u + a; A Qj) 9(u + b
J i
must vanish.
Here €15 e2, ... have any one of 2" possible sets of values. The expression S/^ef1 cannot
i
vanish for every one of these sets ; for, multiplying by e/1, we have then
where fiti, like e^, becomes in turn the symbol associated with every characteristic of the
group, and there are 2*> equations of this form; adding these equations we infer #, = 0,
and, therefore, as^' is arbitrary, we infer that all the coefficients are zero.
510 FIRST APPLICATION [308
Hence it follows that there is at least one of the '2p sets of values for e1} «„, ..., for
which
2cy .9 (u + a; A Q-) 3 (u + b ; A Qj) = 0.
./
When the arguments u + a, u + b are independent, this is impossible; for putting
u + a=U, n + b—V, this is an equation connecting the 2" functions 3(U; AQj) in which
the coefficients are independent of £7(cf. §§ 282, 283, Chap. XV.).
When the arguments u + a, u + b are not independent, this equation is not impossible.
For instance, if flc= -e*1™1 &l, it is easy to verify that
; Qh)3(u; Qh)
and hence the equation does hold when A = 0, a = QQ , 6 = 0, ek= — e*1™'^* •', for all
the values of e1, ..., *,._,, (k + l, ..., fp. For any values of the arguments u + a, u + b
we infer from the reasoning here given that if the functions 9 (u + a ; AQ^S^ + b; AQt)
are connected by a linear equation with coefficients, H^ independent of u, then (i) they
are connected by at least one equation
for one of the 2" sets of values of the quantities fl, f2, ..., and (ii) similarly, since the 2*
functions 3 (u + a; AQi)9(u + b; AQ-) do not all vanish identically, that the coefficients
are connected by at least one equation
309. The result of § 307 is of great generality; we proceed to give
examples of its application (§§ 309 — 313). The simplest, as well as the most
important, case is that in which cr = 0, r=p, and to that we give most
attention (§§ 309—311).
When <r = 0, any two of the functions <£(>, a; A) are connected by a
linear equation, in which the coefficients are independent of u. If v, a, b be
any arguments, and A, B any half-integer characteristics, introducing the
symbol e to put in evidence the fact that 3>(u, a; A) is formed with one
of 2^ possible selections for the symbols e1,...,€p, and so writing <I> (u, a; A,e)
for 4>(w, a; A), we therefore have the fundamental equation
3>(u v A c^^(u>b;B,e)^(a,v-A,e)
By adding the 2? equations of this form* which arise by giving all the
possible sets of values to the fourth roots of unity e^ ..., e^, bearing in mind
that every symbol e{, except e0, = 1, occurs as often with the positive as with
the negative sign, we obtain
(a,v, A, e)
,3>(a,b;B, e)
* Wherein it is assumed that a, b have not such special values that any one of the 2? quanti
ties * (a, 6 ; B, e) vanishes. Of. § 308.
309] OF THE GENERAL THEOREM. 511
whereby the function <f> (u, v ; A) is expressed in terms of 2^ functions
<I> (u, b ; B, e).
By taking, in the formula
3> (u, v; A, e) <& (a, b ; B, e) = ^ (u, b; B, e) <£ (a, v ; 4, e),
or
A E
j X-'1 / \-O
= 2 2 6,6; <*> (»*, 6 ; BPt) ^(a,v; APj).
all the 2^ possible sets of values for elt ..., ep, and adding the results, we
obtain
increasing a and b each by the half-period H^, we have
t; ; ^^Pf) </> (a, b ; 5JKP,-)
«iWf«**«l.#<«tfci BPi)<f>(a)v;
taking R to be all the possible 2^ half-integer characteristics in turn, and
adding the resulting equations we deduce*, putting C = AB,
, b; AC)(j)(a,v; A)
e« i ^ i <f> (u, v • RAPt) <t>(a,b; RA P£)
i R \ ^ J
?(AS\ nilASl., ^ .. , Q~
= Zi I ~ I e • ' " i <p (ii, v ', £>) <p (a, o ; £>0),
s V t/ /
where -4, (7 are arbitrary half-integer characteristics, and S becomes all 2^
possible half-integer characteristics in turn ; for (Ex. ii. § 295), Se7"1-^ ^1 = 2^
it
when P; = 0, and is otherwise zero, while, for any definite characteristic APit
as R becomes all possible characteristics, so does RAP{. The formula can be
simplified by adding the half-period £lc to the argument b; the result is
obtainable directly by taking C= 0 in the formula written.
This agrees with a result previously obtained (§ 292, Chap. XVI.) ; for a
generalisation of it, see below, § 314.
* This equation has been called the Riemann theta formula. Cf. Prym, Untersuchmifien Hber
(lit- Riemaim'telu Thftnfarmcl, Leipzig, 1882.
512 INDICATION OF [310
310. The formula just obtained may be regarded as a particular case of another which
is immediately deducible therefrom. Let (K ) be a group of 2M characteristics formed by
taking all the combinations of p. independent characteristics Klt ..., K^; if A be any
characteristic whatever, we have
according as \A, _ffj|==0 (for i = l, ..., /*), or not; hence, putting (7=0 in the formula
of § 309, and replacing the A of that formula by Kit we deduce
2^ ^ • JT • IT O
2^-M2e"|ji^l^(M, 6; Ki)(f>(a,v, Ari) = 2-'A 2 e"14^1 Se^1^81^ (w, v; S)<j)(a,b; S),
where £ becomes all 22*> characteristics,
= 2 ^2 e7™ 2 c ' * <p (u, V ', S} <f> (a, b ; >S)
,S' i=l
/ 2^ • \
n ~u Tri I j4 1 « iri ^4-R / « Trt 72, Ay \ . / A r>\ i / T ,* />\
=: 2 e 2e (2e /V (%> ^ ! •" ^/ 0 \^*> ^ > •" v>
J{ \i=l /
where R becomes all 22p characteristics,
(a,b; AR\
it
where R extends to all the 22p~'A characteristics for which | R, Ki\ = 0, ..., \ R,
Putting w + Ojg, a + QB for u, a respectively, and replacing AB by C, we obtain
~ ,
** 1 BCL; I • /
e ' J $ (M, v
j=i
here (A") is any group of 2^ characteristics, (Z) is an adjoint group of 22p~'4 characteristics
defined by the conditions \L, K\ = Q (mod. 2), and B, C are arbitrary half-integer
characteristics. The formula of the previous Article is obtained by taking /x = 0. The
formula of the present Article may be regarded as a particular case of that given below
in § 315.
311. The function <j)(u, v; A) is unaffected by the addition of integers
to the half-integer characteristic A ; we may therefore suppose that in the
functions <£ (u, v, APi) which have frequently occurred in the preceding
Articles, the characteristic APi is reduced, all its elements being either 0 or \.
In the applications which now immediately follow (§ 311) it is convenient, to
avoid the explicit appearance of certain fourth roots of unity (cf. Ex. vii.,
p. 469), not to use reduced characteristics. Two, or more, characteristics
which are to be added without reduction will be placed with a comma between
them ; thus A, Pi denotes A + Pt. The characteristics Pt- are still supposed
reduced.
Taking the formula (§ 309)
^ 3> (u, b ; A', e) <£ (a, v; A, e)
-
311] COROLLARIES. 513
where A' replaces the B of § 309, suppose a = b, and put, for
u - b, a + v, a-v, u + v, u-v, a + b, a - b, u + b,
respectively,
U, V, W, U+V, U+W, V+W, 0, U+V+W;
then we obtain
; A)*(U+W; A)
! C')e**(F+ w ; A'>
adding to V and W respectively the half-periods flB, flc, this becomes
**\U, F; A,B][U, W- A,C]
2 2 vMttjW V, W; A', B, C, PJ [U; A', P{] [V;A,B, Pj] [ W; A, C, Pj]
= 2
1 3
2vk8t[V, W; A'.B, C,Pk][0- A',Pk]
k
wherein [U,V;A,B] denotes ^ [U+V- A+B], etc., ^= f 6,, ^ = €it
\^L / V^l /
/ /Q'\ / f\ / '\
etc., and, if £ = £ ) , C' = i(^ ) , P, = i(^' ) , then ^^ sfc are fourth roots of
\P ' \V/ V^r/
unity given by ^^- = e-J»»(^'-h'')(?l.+g(.)) 5jfc = e-JiriO'+y')9Jfct
In connexion with this formula several results may be deduced.
(a) Putting W = - V, A + B = K, A + G = D, A' = D, the formula gives
an expression of *[U+ V; K]*[U-V; D] in terms of the quantities
* [V; KP<], *[U-t DP,], ^[V- DP,], *[0; KP,],
the expre-ssion contains in the denominator only the constants ^ [0 ; KPt],
[0; DP{]; it has been shewn (§ 299) that not all the characteristics KP-,
i can be odd.
Putting further K = Q, we obtain an expression of ^IU+V- 01
*[U-V-, D] in terms of
-, Pt], *[V- P.;], *[U; DP,], ^[F; DP,], ^[0; P,],
Dividing the former result by the latter we obtain an expression for
U+ V- K]/*[U+ V- 0] in terms of theta functions of U &nd Fwith the
characteristics DPt, KP,, P{, the coefficients being combinations of & [0 ; PJ,
^[0; DP,], ^[0; KP{] with numerical quantities. In this expression the
characteristic D is arbitrary ; it may for instance be taken to be zero.
33
514 INDICATION OF [311
The formulae are very remarkable ; replacing, on the right hand, et-ewil^' 'V
by €i, as is clearly allowable, and taking .0 = 0, they are both included in the
following formula (cf. Ex. viii. § 317)
-v; 0]
; P0)
ik'\ !Q '\
where K = 4 7 ) , Pa = i , and the summation in regard to a extends to
2W 2Vga/
all the 2? characteristics, Pa, of the group (P).
It is assumed that the characteristic K is such that the denominator on
the right hand does not vanish for any one of the 2? sets of values for the
quantities ea . For instance the case when K is one of the characteristics of
the group (P), other than zero, is excluded (cf. § 308).
Ex. i. For p = l, if P denote any one of the half-integer characteristics other than
zero,
[3* (u) 32 (v) + 3*p (u) 4 („)] 52 (0) - [32 (u) 4 (v) +e™ I p 1 ^ (it) 32 (*)] 4 (0)
where 5 (w), ^P(w) denote 3 (u; 0), 5 (u; P), etc.
.£&. ii. By putting, in case p = 2,
deduce from the formula of the text that
4^2 (0) $01 (°) 502 (" + «') ^5 (v ~u')= 2
^1 > ^2
wherein £,= +1, f2= +1, and
A', B', C", I? denoting the same functions of the arguments u'.
Hence obtain the formula given at the bottom of page 457 of this volume.
08) Putting B= C, V=W=0,A' = A, we obtain
tij [U; A, B, B, P{][U; AP<] [0 ; A, B,
o; A,B,B,Pk][0; A, Pk]
k
which shews that the square of any theta function is expressible as a linear
function of the squares of the theta functions with the characteristics forming
the Gopel system (^1P). We omit the proof that these 2? squares,
^(U; APi), are not in general connected* by any linear relation in which
the coefficients are independent of U.
* Cf. the concluding remark of § 308, § 291, Ex. iv. and § 283.
311] COROLLARIES. 515
Ex. For p = 2 obtain the formula
where 52 = 52 (0), etc.
(7) There is however a biquadratic relation connecting the functions
£ (u ; APi) provided p be greater than 1. In the formula (§ 309)
(a-b; A, Pt)
(a-v; A, 1\\
supposing the characteristic A to be chosen so that all the characteristics
APt are even, as is possible (§ 299) by taking A suitably, substitute for
u + v, u-v, a + b, a — b, u + b, u—b, a + v, a—v
respectively
u + v + w, u-v, a + b + w, a-b, u + b + w, u - b, a + v+w, a - v;
then, putting a = b = 0, we have
n-v- A, Pi)*(u + v + w ; A, P<)
(v + w; A, P,);
herein put w = nPi, v = u + £LPn, where P,, P2 are two of the characteristics
belonging to the basis Plt ..., Pp of the group (P) ; then we obtain
/PP
l)r<iJM»<tii 4, Pf)*(«; ^, P,, PW*; ^^2> A-)^(«; A, plt p., PA
t /
Now every characteristic of the group (P) can be given in one of the forms
Qg, Q*Pi, QsPz, Q«PiP2, where Qs becomes in turn all the characteristics of
a group (Q) of 2^~2 characteristics ; putting
we immediately find
* (« ; Q.) = * (« ; Q., P,) = *(*;&. A) = ^ (*« ; Q-, A, P9) ;
hence the equation just obtained can be written
u
where Rm has the four values 0, P,, P,,, P, + P.,.
Again, if in the formula (§ 309)
33—2
516 COROLLARIES. [311
we add to u the half period Op , we obtain, after putting u = v, a = b = 0, the
result
,0; A,e)'
where
By substitution of the value of ^ (2u ; A, Pk) given by this formula, in
the formula above, there results the biquadratic relation* connecting the
functions ^ ('ii ; A PI).
(8) As an indication of another set of formulae, which are interesting as
direct generalizations of the formulae for the elliptic function $(u), the
following may also be given. Let
a a
o — A,j ~ r • • • ~r A^ ,5 ,
dvl r ovp
where \, ...,\p are undetermined quantities, SS- (v) = & (v*), 8*$ (v) = W (v),
and let
+W(v; A)-
then, differentiating the formula
(u, b; A, e) 3> (a, v ; A, e)
'
twice in regard to v, and afterwards putting v = 0 and b = 0, we obtain
wherein
(*j APk)
the 2^ quantities Ct being independent of M and of a. By this formula the
function %>(u', A) is expressed linearly by the squares of 2P theta quotients
(cf. Chap. XI. § 217).
* Frobenius, Crelle, LXXXIX. (1880), p. 204. The general Gopel biquadratic relation has also
been obtained algebraically (for Riemann theta functions) by Brioschi, Annal. d. Mat., 2a Ser. ,
t. x. (1880—1882).
312] SECOND APPLICATION OF THE GENERAL THEOREM. 517
312. These propositions (§§ 309 — 311) are corollaries from the fact that
the functions Q(u, a; A, e) are linearly expressible by 2*'~r of them; we
have considered the case r=pa,t great length, on account of its importance.
Passing now to the case r = p — 1, there is a linear relation connecting
any three of the functions
<f> (u, a ; A, e) = 2S (Pf] ei* (u + a- AP{) *(u-a; APt).
i = l \-™- /
There is one case in which we can immediately determine the coefficients in
this relation ; we have <r = p -r=l, 22(T = 4 ; there are thus four character
istics A, whereof three are even and one odd, which are such that all the
2P-1 characteristics (AP) are of the same character. Taking the single case
in which these are all odd, we have
4>(M, a; A, e) = -<£(«, u; A, e), and $> (a, a; A, e) = 0;
hence, if, in the existing relation
X4> (u, a; A,e) + n$>(u,b; A, e) + i/<X> (u, c; A,e) = 0,
wherein X,, p, v are independent of u, we put u = a, we infer
yu : v = <5> (c, a ; A, e) : <b (a, b ; A, e) ;
thus the relation is
, c- A,€)®(u,a; A, e) + 3> (c, a ; A,€)®(u,b; A, e)
+ 4> (a, 6; A, e) <!>(«, c; A,e) = 0,
or
2P-1
V
where
-c-, AP})
-a; AP,)
Adding together all the equations thus obtainable, by taking all the
possible sets of values for the fourth roots of unity ely ..., ep-lt we obtain
For instance, when /> = !, this is the so-called equation of three terms, from which all
relations connecting the elliptic functions can be derived. When p = 2, it is an equation
of six terms and there are fifteen such equations, all expressed by
'•;A)$(b-c',A)
A and B being any two odd characteristics*.
* Cf. Frobenius, Crelle, xcvi. (1884), p. 107.
518 THIRD APPLICATION OF THE GENEKAL THEOREM. [313
313. Taking next the case r=p — 2, every 22 + 1, or 5, functions
<£> (w, a; .4, e) are connected by a linear relation. In this case there are
sixteen characteristics A such that all the 2p~2 characteristics (AP) are of
the same character, six of them being odd. Denoting the six odd character
istics in any order by A1} ..., As, and an even characteristic by A, there is an
equation of the form
\^ (u, a; Alte) + X23> (u, a ; A2, e) + \3<& (u, a; A3, e)
= 4> (u, a ; J.4, e) + X3> (w, a ; A, e) ;
putting herein u = a, this equation reduces to \<I>(a, a; J., e) = 0, so that
A, = 0. The other coefficients can also be determined ; for, if C = A2A3, we
have (§ 306, Ex. i.),
4> (M + flc, a ; A, e) = e*(«-. O (^a/3) <X> (ti, a ; 44.A,, e) ;
putting therefore for u, in the equation above, the value a + Hc, where
C= A2A3, and recalling (§ 303) that A^A^As, A4A2A3 are even characteristics,
we infer
X, * <S> (a, a • A,A2AS, e) = ** 4> (a, a ; A,A,A3, e).
Proceeding similarly with the characteristics A3Alf AjA2 in turn, instead of
A2A3, we finally obtain
3> (a, a ; AtAM <$> (u, a- A,) + •* 3> (a, a ; A4A3A,) & (u, a ; A,)
\-d.2-^4'
> (a, a; A.A.A,) $> (u, a ;. A3) = $>(a,a; A.A^A,) <& (u, a ; At),
where, for greater brevity, the e is omitted in the sign of the function 3>
(cf. Ex. viii., § 289).
Ex. For p = 2, deduce the result
*MSM (2-t;) 502 (u + v) \, (u-v)- 403303 (2 v) $u (u + v)9M(u-v) + 523^23 (Zv)^(u+ v} S0i (u-v)
where ^34 = ^34(0), etc. When v = Q this is an equation connecting the squares of 302(«),
3-24 (tt), ^04 (»), ^1 («)-
314. The results of §§ 309, 310 are capable of a generalization, obtainable by a repeti
tion of the argument there employed.
A group of 2fc pairwise syzygetic characteristics may be considered as arising by the
composition of two such groups. Take k, = r + s, characteristics Ply ..., Pr, Qi, •••> Qs,
every two of which are syzygetic ; form the groups
(P) = 0, /j, ..., Pr, PI"Z, '•', «j«»*n •••
respectively of 2>- and 2s characteristics ; the 2>- + 8 combinations Riyj = PiQj form a group
(/£) of 2r + * pairwise syzygetic characteristics; for distinctness the fourth roots of unity
314] DEDUCTION OF A FURTHER RESULT. 519
associated respectively with 1\, ..., Pr, Qlt ..., Qt, may be denoted by *j, ..., fr, f,, ..., („ ;
then with /W, Qj,jlt Ri,j will be associated the respective quantities
(Pi\ ,, (Qt\ „
«<.<, = «««, (pj , &,* = &<* (QJ , ««-«
thus if J be any characteristic
- . .
A "\ A « Q, - ; • *
Therefore, using the symbol ¥ for a sum extending to the whole group (PQ\
* (u, a ; A, E)= 2 ' E{ f $ (u + a ; AR^) 3(u-a; ARU]
; AQJPi)9(u-a; AQ.P,)
where * denotes a sum extending to the 2r terms corresponding to the characteristics of
the group (7*).
By the theorem of § 307 the functions obtainable from *• (?*, a ; A, E) by taking
different values of a and A, and the same group (PQ), are linearly expressible by
2P-r-«=2<r-« Of them, if v=p — r, with coefficients independent of u. The 2s functions
* (u, a ; AQjy e), obtained by varying a and Qit are themselves expressible by 2°" of them.
Thus, taking r+s=p, or s = cr, we have
* (M, w ; J, J£) * (a, 6 ; ^1, ^) =* (u, b ; 4, #) V (a, v; A, E)
or
.2 (^
= s
taking for £\, ..., f, all the possible 2* values, and adding the 2* equations of this form,
we obtain
2e"lfcl*(M, v; 4Q,,f)*(a,6; 4^, «)=2 «H|*'#(%6j .!§>, e) * (a, » ; 4&, e).
J=i j=i
Suppose now that A1,...,A^ are the 2217 characteristics satisfying the r relations
| X, Pi | = | PJ |, (mod. 2), and let (7m=^1^4m ; then | Cm) Pi | = 0 ; hence, by the formulae of
§ 306, Ex. i., adding the half period QCm to u and 6, and dividing by the factor e77*1*7"" Al,
we have
taking, here, all the 22<r values of (7m in turn, and adding the equations, noticing that
is zero because Qj is not a characteristic of the group (/'), except for the special value
Qj=0, when its value is 22<r (§ 300), we derive the formula
2*r * (M, i ; J, 0 * («, » ; '1,0=2 S c'f ic»«°> ' « (M, v ; .IC',,,^, f) * («, b ; .K',,,^, e) ;
J-=l m=l
520 DEDUCTION OF A FURTHER RESULT. [314
now, as already remarked (§ 298, Ex.), if a characteristic S which is syzygetic with
every characteristic of the group (P) be added to each of the 22<7 characteristics Alt ..., Ak
the result is another set of 22a characteristics satisfying the same congruences, | X, I\ = \ 1\\ ,
as the set A1} ..., A^, and incongruent mod. (P) ; thus, taking a fixed value of j, we have
CmQj=CnPi, where, as C,n takes its 22<T values, Cn also takes the same values in another
order, and Pi varies with m. Hence (Ex. iii. § 306) we have
^KW *(w,t,; ACmQJt e)* (0,6; ACmQj, f) = enilc*pi l * (u, v- ACnPiti) * (a, b; ACnPit «),
= e^c«<*(u,v; ACn,()*(a,b; ACn,t),
and
and therefore, finally, dividing by a factor 2' (there being 2<r characteristics in ($)), we
have
2'2<T
2** (w,6; A, «)*(«, v; ^,0= 2 e*1^* '*(*,«; 4J^m, e)*(a, 6; ^J^m, e).
m=l
When 0-=^, this becomes the formula of § 309. We infer that the functions
<J>(M, a; -4, e) are connected by the same relations as the functions of the form
; A) 3 (u-a; A] when the number of variables (in the latter functions) is <r.
Ex. Prove that, with the notation of the text,
f *(a,b;A,E)
315. The formula of the last Article is capable of a further generalization. Let (R) be
a group of 2^ characteristics, formed with Jtlt ..., R^ as basis, which satisfy the conditions
R P =0 /f P \ = 0
Jt, 1 j — V, . . . , ./»,, 1 ,. | — U.
Thus (P) is a sub-group of (R) ; the group (If) consists of (P), together with groups (RP),
whereof the characteristics R form a group of 2*~r characteristics, whose constituents are
incongruent for the modulus (P). The basis of this sub-group of 2^~J' characteristics will
be denoted by R1, ..., R,j._r. The total number of characteristics satisfying the prescribed
conditions is 22p~r; thus p^2p — r, and, when /z<2jo-r the given conditions are not
enough to ensure that a characteristic belongs to the group (R).
Then, if F, G be arbitrary characteristics, and Rt become in turn all the characteristics
of a group of 2A*~r characteristics of the group (R) which are incongruent mod. (P), we
have
I
* («, 6; GF&i, t) * (a, v ; GRt, e)
SJM-r
where (7m = A^Am. Since | ^, P =0, the constituents of the set RiCm, where /?» is a fixed
characteristic and m=l, 2, ..., 22(T, are in some order congruent (mod. (P)) to the con
stituents of the set Cm ; hence (§ 306, Ex. iii.) the series is equal to
M, v ; 6'<7m, e) * (a, b ; (70^, c),
*ilj $ (M) v; (yc^, e) * (a, 6; (?(7m, e) ;
316] A GENERAL ADDITION FORMULA. 521
<jy-r
now 2 enl ' * is zero, \mless \L, lii\=0 (mod. 2) for every characteristic Ii^ in which
i=l
case its value is 2'4~r ; thus the series is equal to
where Sm satisfies the conditions involved in | Sm, Rf =0, FGCm=Smj namely the con
ditions
\Sm, R, =0, ..., \Sm, ^_r|sO, FGSm, Pl =0, ..., \FGSm, Pr\ = 0 ',
the number of characteristics satisfying these /* conditions is 22^~^; the number of these
which are incongruent for the modulus (P) is 22P~'*~r=22o'+r~''i.
Suppose now that \FG, Pj =0, ..., | FG, Pr | = 0 ; then the characteristics Sm con
stitute a group satisfying the conditions | Sm, R =0, where R becomes in turn all the 2^
characteristics of the group (R). The group (S) of the characteristics Sm may be obtained
by combining the characteristics of the group (P) with the characteristics of a group of
2 f-f. r characteristics which also satisfy these conditions and are incongruent for the
modulus (P) ; putting fj. = r + p, we have therefore*
•i, 0
»'*(«,*; FSm,t)*(a,b; FSm, e).
In this equation each of Rit Sm represents the characteristics, respectively of the
groups (R), (S), which are incongruent mod. (P). But it is easy to see (§ 306, Ex. iii.)
that we may also regard Rt, Sm as becoming equal to all the characteristics, respectively,
of the groups (R), (S}.
316. We have shewn in Chap. XV. (§ 286, Ex. i.) that a certain addition
formula can be obtained for the cases p= 1, 2, 3 by the application of one
rule. We give now a generalization of that rule, which furnishes results for
any value of p.
Suppose that among the 22<r characteristics Alt A2, ..., A^ which, for any
Gopel system (P) of 2r characteristics, satisfy the conditions
we have k + 1 = 2-+ 1 characteristics B,, ...,Bk, B, of which B is even, which
are such that, when i is not equal to j, BBiBj is an odd characteristic ; as
follows from § 302 of this chapter, and § 286, Ex. i., Chap. XV., this is
certainly possible when o- = 1, or 2, or 3 ; and, since
\BBiBj,P\= B,P\ + \Bi,P + ^,Pj = |P|,
* The formula is given by Frobenius, Crelle, xcvi. p. 95, being there obtained from the
formula of § 310, which is a particular case of it. The formula is generalised by Brauninuhl to
tlieta functions whose characteristics are n-th parts of integers in Math. Annal. xxxvn. (1890),
p. 98. The formula includes previous formulae of this chapter.
522 A GENERAL ADDITION FORMULA. [316
the characteristics BBiBj will be among the set A1} ..., A^, so that all
characteristics congruent to BBiBj (mod. (P)) are also odd. Then by § 307
there exists an equation of the form*
k
, c; B,e) = 2 \m$>(u, a; Bm, e),
m=l
wherein the coefficients \, \lt ..., \k, are independent of u. Put in this
equation u = a + O,BB. ; then we infer (§ 306, Ex. i.)
X<J> (a, c ; BI, e) = X;<£ (a, a ; -B, e) ;
hence we have
k
$>(a,a; B, e) 3> (M, c ; B, e) = 2 e^\^Bm\ <|> (a> c - sm, e) <l> (M, a ; Bm, e),
m=l
which is the formula in question f.
Adding the 2r equations obtainable from this formula by taking the
different sets of values for the fourth roots of unity els ..., er, there results
2 e-f I ^ l^-o (#?<)= 2 2 e-l
i=l m=l i=l
where
= * (0 ; BPi) * (2o ; &P<) ^ (u + c ; JBP*) ^ (u - c ;
= * (a + c ; BJPi) * (a - c ; 5^) ^ (u + a ; 5w
Herein we may replace the arguments
2a, u + c, u — c, a + c, a — c, u + a, u — a
respectively by
U, V, W, i(U+ V- W\ ±(U- V+ W), ±(U+ V+ W), i(- U+ V+ W),
and thence, in case p = 2, or p = 3, obtain the formula of Ex. xi., § 286,
Chap. XV.
Or we may put a — 0, and so obtain
2r
2e«lpil^(0; BPi)*(u + c; BPi)*(u-c; BP{}
1=1
= £ | e«i\Bm,BPi\^(u. ^p.)^^. Bmp.}.
m=\ i=\
Other developments are clearly possible, as in § 286, Chap. XV.
Ex. When ar=l there are three even Gopel systems, and one odd; let (BP\ (B^P),
(B2P) be the three even Gopel systems; then we have
* (a, a ; B, c) * (u, c; B, e)
= e*i|M''*(a, c; B,, «) * (a, a; Blt f} + eni\BB*\ * (a, c; 52, c) * (M, a; £2, e),
* We may, if we wish, take, instead of the characteristic B on the left hand, any characteristic
A such that | A, Pf \ = \ P( | , (i = 1, ... , 2'').
t For similar results, cf. Frobenius, Crelle, LXXXIX. (1880), pp. 219, 220, and Noether, Math.
Annal. xvi. (1880), p. 327.
317] EXAMPLES. 523
where * («, a ; B, t) consists of 2""1 terms ; for instance when p= 1 we obtain
3(0; B)3(2a; B)3(u + c; B)9(u-e; B)
r, Bl)3(a-c; Bl)9(u + a; B1)3(u-a; Bj)
B2)3(u-a; B2).
317. Ex. i. If P be a fixed characteristic and <1f(u; A) denote the function
3(u; A)3(u; A + P), prove that
>ty(ii- A\
- . * \ «• > -" ;,
and
'•, B+Q).
Hence, if B^ ..., Bk, Bbe ^ + 1=2P-1 + 1 characteristics each satisfying the condition
I JT, P\= | P\, such that, when i is not equal to j, BB^B, is odd, we have (§ 307) an
equation
2P-1
A¥(«; .4)= 2 \mV(u; Bm),
where ^1 is any other even characteristic such that | A, P\ = \ P | ; putting u = QB + a#., we
obtain
therefore
fia-. ii. Obtain applications of the formula of Ex. i. when p = 2, 3, 4; 'in these cases
"•» =P-1> =1» 2> 3 respectively, so that we know how to choose the characteristics
B11...tBk,B (Ex. i., § 286, Chap. XV., and § 302 of this Chap.).
Ex. iii. From the formula (§ 309)
5(M + 6; A)3(u-b; A)$(a + v; A)3(a-v; A)
= -2e*i\AR\9(u + v, R)B(u-v; R)9(a + b; Ji)S(a-b; R),
by putting a + QP for a, and b = v=0, we deduce
52 (tt; A) 3* (a; JP) = 2^2 e^1^' f /' } 3* (u- R)9*(a-, PR),
it \aLK/
where A, I1 are any half-integer characteristics and R becomes all the 22" half-integer
characteristics in turn ; putting RP for R we also have, from this equation,
; If);
therefore
; AP)
The values of R may be divided into two sets, according as \R,P\ + \P\ = l (mod. 2),
or =0; for the values of the former set the corresponding terms vanish; the values of R
for which \R,P\ + \P\ = Q (mod. 2) may be either odd or even; for the odd values the
zero values of the corresponding theta functions are zero ; there remain then (§ 299) only
2. 2" -2(2"-' + 1) terms on the right hand corresponding to values of R which satisfy the
524 EXAMPLES. [317
conditions | R \ = \ RP \ = 0 (mod. 2) ; these values are divisible into pairs denoted by
R = E, R = EP; for such values \ + e^R' p l+"lpl = 2, and
i\AE\( f\,eni\AEP\( 1" \
\AE)+ \AEP)
,*i\AE\ ( P \ r-i.jd\AB,P\-in\AJB\( P r, , ri\A,
=< AE L
P\ r, , ri\
AE) L
thiis, provided | A, P \ + 1 P \ = 0 (mod. 2),
(; EP), (i),
wherein 32(; -4) denotes 32(0; J), etc., and, on the right hand there are 2"~2 (2P~
terms corresponding to values of E for which | E\ = \ EP | = 0 (mod. 2), only one of the two
values, E, EP, satisfying these conditions being taken.
Putting P=0, u = a, in the second equation of this example, we deduce in order
S*(M; A) = Z-^e'^AR^*(u', R); 3*(u; AP} = Z~^e^APR\ & (u; R);
R R
so that, by addition,
$*(u; A) + eiri\A'Pl3i(u; ^P)=2-"S«"<l^Jl|[H-eir*|p|+iri|Ji"p|]5*(«; R);
R
thus, as before,
; EP)}, (ii).
Ex. iv. Taking p = 2, let (P) = 0, Px, P2, P^P-,, be a Gopel group of even charac
teristics*; let Bl, S2, B^BI be such characteristics (§ 297) that the Gopel systems
(P), (B1P\ (BZP}, (J31B2f} constitute all the sixteen characteristics; each of the systems
(B^P), (B2P), (BlBi>P) contains two odd characteristics and two even characteristics.
Then, in the formulae (i), (ii) of Ex. iii., if P denote any one of the three characteristics
P15 P2J PlP2t the conditions for the characteristics E are E, P\= P| = 0, 1^1 = 0; the
2. 2^~2(2p~1 + l), =6, solutions of these conditions must consist of 0, Q, B and P, QP, BP,
where Q is defined by the condition that the characteristics 0, Q, P, QP constitute the
group (P), and B is a certain even characteristic chosen from one of the systems (Bl P),
(B2P), (B1B2P). Hence, when P=Pi, we may, without loss of generality, take for the
2P-2 (2p~1 + l) = 3 values of E which give rise to different terms in the series (i), (ii), the
values 0, P2, Bl; similarly, when P=P2> we have, for the values of E, E=0, P1? B2; and
when P=P1P2, E=0, P15 B^; taking A to be respectively t Bly B2, B^ in these
cases, we obtain the six equations
* There are six such groups (Ex. iv. § 289).
f We easily find | BlB%Pl \ = \ B^P* = - | BJt* j . Thus the case when B^ is odd is
included by writing B1P1 in place of 2ix.
317] EXAMPLES. 525
wherein ^.^i»/ _ _ i. These formulae express the zero values of
all the even theta functions in terms of the four£(; 0), B(; PJ, $(; Pz), 3(; P^P^.
Thus for instance they can be expressed in terms of ^5,^34, S12, $0; the equations have
been given in Ex. iii., § 289, Chap. XV.
Ex. v. We have in Chap. XVI. (§291) obtained the formula
S(u-v,
where t represents a set of p integers, each either 0 or 1, and has therefore 2» values.
Suppose now that q, r represent the same half-integer characteristic, =i( ) +| ( /. ) ,
\c / Vp«/
= C+Ka, say; then we immediately find
*.[«' *^£+o>[-' *«3J#KT' t' y •[- *r|.
where t'c' denotes the row of p integers, each either 0 or 1, which are given by (c'<f)i = fj+cj
(mod. 2); herein the factor e"wc' S^ •» is independent of ka. For Ka we take now, in
turn, the constituents
0, Klt A2, ..., Kp, KlK.i, ..., K^K^K^ ...
of a Go pel set of 2p characteristics, in which
0,0,0, ...\ 1/0,0,0,..A /O, ...,0,0\
=
then denoting $[u + v; CKJ $[u-v; CKa] by [CKa], we obtain 2» equations which are all
included in the equation
wherein *=2P, e/, ..., es' represent the different values of «', and ,/ is a matrix wherein the
/3-th element of the a-th row is ^ \u ; -f&C \.
The 2p various values of c'^c', for an assigned value of c', are, in general in a different
order, the same as the various values of t'o ; we may suppose the order of the columns of
t7 to be so altered that the various values of e'^c' become the values of e'« in an assigned
order, the order of the elements &™c' ^ \v ; 5 * 1 , . . . , «"•'"«' ^ \v ; ^ ** being correspond
ingly altered. When this is done the matrix J is independent of the characteristic C.
Now it is possible to choose 2" characteristics (7, say Clt ..., Cg such that the Gopel
systems (CtK) give, together, all the 2" possible characteristics ; then the 2? equations
obtainable from that just written by replacing C in turn by Clt ..., C,, are all included,
using the notation of matrices, in the one equation*
wherein £'a denotes a row of jt? integers, each either 0 or 1, and has 2" values. In each
matrix the element written down is the ;3-th element of the a-th row.
* We can obviously obtain a more general equation by taking 22" different sets of arguments,
the general element of the matrix on the left hand being 3-[>(a) +v^} ; CaKfi]$[uw -v®] ; CaKft].
Cf. Chap. XV. § 291, Ex. v., and Caspary, Crelle, xcvi. (1884), pp. 182, 324; Frobenius, Crelle,
xcvi. (1884), p. 100. Also Weierstrass, Sitzuni/abfi: der Ak. <l. Wins, zn Herlin, 1882, i.— xxvi
p. 500.
526 EXAMPLES. [317
Ex. vi. If in Ex. v., jo = 2, and the group (A') consists of the characteristics
oi
while the characteristics C consist of
/00\ , /10\
* (oo) ' 2 (oo) '
and the values of f are, in order,
(0,0), (0,1), (1,0), (1,1),
shew that the sixteen equations expressed by the final equation of Ex. v. are equivalent to
= ( «4, 0.5, — a.2, "i ) ( ft? —ft? ft) ft)
oon rion ron
iiJ' [_ooj' L10J'
O
n
roii
' LHJ
oi
oi
o1
io
00
— ao , a
o , a4 , nj , a2
a0, a,, —i
«2> -°1) a4) «3
Q C) Q Q
P-2» P3' ft> "ft
-ft) ft) ft) ft
ft) ft) "ft) ft
wherein, on the left hand, denotes S u + v; ?[-,-.} ^ u-v; |(n J L etc., and
the right hand,
--*.[«! *Q]. *-«.[»!
/31? /32, ft, ft being respectively the same theta functions with the argument v.
Now if A, B denote respectively the first and second matrices on the right hand, the
linear equations
(#1. ^2> #3> 2/4) = ^ (#11 -^2> '^3> -r4)> (-^I, #2. 'r3> ^4) = -B(«;l» «2> %» ^4)
are immediately seen to lead to the results
V+ V+V+ V=(ft2+ft2+ft2+ft2) (V+^+V+O ;
hence if the^-th element of the j'-th row of the compound matrix J.B, which is the matrix
on the left-hand side of the equation, be denoted by y?. . , we have
i1t«"i*l*»i^rn.'"^ (r**' r' *=1» 2' 3» 4)'
and these equations lead to
A<*-A*^A^'v^
Denoting , , by [a^g] , [a^J , etc., as in the table of § 204, and inter
changing the second and third rows of the matrix on the left-hand side, we may express
the result by saying that the matrix
K^L KcJi -|>i<| , [aj
-[cc,] , -[cc2] , -[0^2], [0]
gives an orthogonal linear substitution of four variables*.
* An algebraic proof may be given ; cf. Brioschi, Ann. d. Mat. xiv.
317] A MULTIPLICATION FORMULA. 527
Ex. vii. Deduce from § 309 that
AP^^(v; Al>S\
— - ,
e ifa * (,U > OJTf}
a.
where /»<, Pa are characteristics of a Gopel group (P\ of 2» characteristics. Infer that, if
n be any positive integer, and A Pi be an even characteristic, 3(nv; APj) is expressible as an
integral polynomial of order n2 in the W functions 9(v; AP ).
Ex. viii. If K = | ( ) , P. = | (q a ) , deduce from § 309, putting
W \?a/
a = b = u- U=v— V=\Qk,
that
r, -F),
where
x (u, v] = 2fae~ "'»« 5 (« ; /r+ P0) 3 (v ;
[318
CHAPTER XVIII.
TRANSFORMATION OF PERIODS, ESPECIALLY LINEAR TRANSFORMATION.
318. IN the foregoing portion* of the present volume, the fundamental
algebraic equation has been studied with the help of a Riemann surface.
Much of the definiteness of the theory depends upon the adoption of a
specific mode of dissecting the surface by means of period loops ; for instance
this is the case for the normal integrals, and their periods, and consequently
also for the theta functions, which were defined in terms of the periods
Tij of the normal integrals of the first kind; it is also the case for the
places ml,...,mp of § 179 (Chap. X.), upon which the theory of the
vanishing of the theta functions depends. The question then arises ; if we
adopt a different set of period loops as fundamental, how is the theory
modified, and, in particular, what is the relation between the new theta
functions obtained, and the original functions ? We have given a geometrical
method (§ 183, Chap. X.) of determining the places ml, ...,mp from the
place m, from which it appears that they cannot have more than a finite
number of positions when m is given, and coresidual places are reckoned
equivalent; the enquiry then suggests itself; can they take all these possible
positions by a suitable choice of period loops, or is one of these essentially
different from the others ? The answers to such questions as these are to be
sought from the theory of the present chapter.
There is another enquiry, not directly related to the Riemann surface,
but arising in connexion with the analytical theory of the theta functions.
Taking p independent variables ul} ..., itp, and associating with them, in
accordance with the suggestion of §§ 138 — 140 (cf. § 284), the matrices
2&), 2&/, 2rj, 2?/, we are thence able, with the help of the resulting equations
2//W = TTI, 2ha>' = 6, i) = 2a&>, rf = 2aa>' — h,
to formulate a theta function. But it is manifest that this procedure makes
an unsymmetrical use of the columns of periods arising respectively from
the matrices co and o>' ; and it becomes a problem to enquire whether this
* References to the literature dealing with transformation are given at the beginning of
Chap. XX.
319] THE GENERAL SYSTEM OF PERIOD LOOPS. 529
want of symmetry can be removed ; and more generally to enquire what
general linear functions of the original 2p columns of periods, with integral
coefficients, can be formed to replace the original columns of periods; and, if
theta functions be formed with the new periods, as with the original ones,
to investigate the expression of the new theta functions in terms of the
original ones.
So far as the theta functions are concerned, it will appear that the
theory of the transformation of periods, and of characteristics, includes the
consideration of the effect of a modification of the period loops of a Riemann
surface ; for that reason we give in this chapter the fundamental equations
for the transformation of the periods and characteristic of a theta function,
when the coefficients of transformation are integers ; but the main object
of this chapter is to deal with the transformation of the period loops on a
Riemann surface. The analytical theory of the expression of the transformed
theta functions in terms of the original functions is considered in the two
following chapters.
In virtue of the algebraical representation which is possible for quotients
of Riemann theta functions (as exemplified in Chap. XI.), the theory of
the expression of the transformed theta functions in terms of the original
functions, includes a theory of the algebraical transformation of the funda
mental algebraical equation associated with a Riemann surface ; it is known
what success was achieved by Jacobi, from this point of view, in the case of
elliptic functions ; and some of the earliest contributions to the general
theory of transformation of theta functions approach the matter from that
side*. We deal briefly with particular results of this algebraical theory in
Chap. XXII.
319. Take any undissected Riemann surface associated with a funda
mental algebraic equation of deficiency p. The most general set of 2p
period loops may be constructed as follows :
Draw on the surface any closed curve whatever, not intersecting itself,
which is such that if the surface were cut along this curve it would not be
divided into two pieces ; of the two possible directions in which this curve
can be described, choose either, and call it the positive direction ; call the
side of the curve which is on the left hand when the curve is described
positively, the left side ; this curve is the period loop (A^ ; starting now
from any point on the left side of (A^, a curve can be drawn on the surface,
which, without cutting itself, or the curve (A^, and without dividing the
surface, ends at the point of the curve (AJ at which it began, but on the
right side of (AJ ; this is the loop (BJ, and the direction in which it has
* See, in particular, Richelot, Crelle, xvi. (1837), De transformatione...integralium Abelian-
orum primi ordinis ; in the papers of Konigsberger, Crelle, nx.iv., LXV., LXVII., some of the
algebraical results of Richelot are obtained by means of the transformation of theta functions.
B. 34
530 THE GENERAL SYSTEM OF PERIOD LOOPS. [319
been described is its positive direction ; its left side is that on the left hand
in the positive description of it. The period associated with the loop (^i),
of any Abelian integral, is the constant whereby the value of the integral
on the left side of (Aj) exceeds the value on the right side, and is equal to
the value obtained by taking the integral along the loop (B^ in the negative
direction, from the end of the loop (BJ to its beginning. The period
associated with the loop (BJ is similarly the excess of the value of the
integral on the left side of the loop (B^ over its value on the right side, and
may be obtained by taking the integral round the loop (A^) in the positive
direction, from the right side of the loop (BJ to the left side. These periods
may be denoted respectively by Ox and fl/.
320. It is useful further to remark that there is no essential reason why what we have
called the loops (Aj), (B^j should not be called respectively the loops [5J and [JJ. If
this be done, and the positive direction of the (original) loop (Z?t) be preserved, the
convention as to the relation of the directions of the loops [A{\, [B^\ will necessitate a
reversal of the convention as to the positive direction of the (original) loop (Aj). If the
periods associated with the (new) loops [A^, [B^\ be respectively denoted by [Q] and [Q'],
we have, therefore, the equations
These equations represent a process — of interchange of the loops (AJ, (BJ, with retention
of the direction of (BJ — which may be repeated. The repetition gives equations which we
may denote by
{Q} = [O'J = - O, {&'} = - [Q] = - Q',
and the two processes are together equivalent to reversing the direction of loop (Aj)t and
(therefore) of the loop (Bj). The convention that the loop (BJ shall begin from the left
side of the loop (Aj) is not necessary for the purpose of the dissection of the surface into a
simply connected surface ; but it affords a convenient way of specifying the necessary
condition for the convergence of the series defining the theta functions.
321. The pair of loops (AJ, (Bj) being drawn, the successive pairs
(A2), (.B2), ..., (Ap), (Bp) are then to be drawn in accordance with precisely
similar conventions — the additional convention being made that neither
loop of any pair is to cross any one of the previously drawn loops. If
the Riemann surface be cut along these 2p loops it will become a p-ply
connected surface, with p closed boundary curves. It may be further
dissected into a simply connected surface by means of (p — 1) further cuts
((7j), ..., (C^-j), taken so as to reduce the boundary to one continuous closed
curve.
Upon the p-p\y connected surface formed by cutting the original surface
along the loops (AJ, (B^, ..., (Ap), (Bp\ the Riemann integrals of the first
and second kind are single- valued. In particular if Wl} ..., Wp be a set of
linearly independent integrals of the first kind defined by the conditions
that the periods of Wr at the loops (A^, ..., (Ap) are all zero, except that at
322] CHANGE FROM ONE SYSTEM TO ANY OTHER. 531
(Ar\ which is 1, and if rr,g be the period of Wr at the loop (Bs\ the imaginary
part of the quadratic form
Tunj2 + ...... + 2T12n1n2 + ...... + TpflPp
is necessarily positive* for real values of n1, ..., np. This statement remains
true when, for each of the p pairs, the loops (Ar), (Br) are interchanged,
with e.g. the retention of the direction of (Br) and a consequent change in the
sign of the period associated with (Ar\ as explained above (§ 320) ; if the
loops (Ar), (Br) be interchanged without the change in the sign of the period
associated with (Ar), the imaginary part of the corresponding quadratic
form is negative*}-.
322. In addition now to such a general system of period loops as has
been described, imagine another system of loops, which for distinctness we
shall call the original system ; the loops of the original system may be
denoted by (ar), (br) and the periods of any integral, ui} associated therewith,
by 2ft>;ir, 2(o'i>r; the general system of period loops is denoted by (Ar), (Br),
and the periods associated therewith by [2^ .,.], [%w'it r]. For the values of
the integral ui} the circuit of the loop (Br), in the negative direction, from
the right to the left side of the loop (Af), is equivalent to a certain number,
sayj to a.j>r> of circuits of. the loop (bj) in the negative direction, together
with a certain number, say a 'j> r, of circuits of the loop (a,-) in the positive
direction (r,j=I, 2, . .., p}] hence we have
p
[a>it r] = 2 (&>;, j «/, r + G>\ j*'j,r\ (r — 1 , 2, . . . , p ) ;
j=i
similarly we have equations which we write in the form
K & r + a>\ jfi'j, ,), (r = 1, 2, . . . , p),
the interpretation of the integers /8j>r, j3'jjr being similar to that of the
integers «,-_,., afj>r.
Thus, if uly ...,up denote p linearly independent integrals of the first
kind, and the matrices of their periods for the original system of period
loops be denoted by 2&>, 2&>', and for the general system of period loops by
[2<a], [2ft)'], we have
[o>] = wo. + a/a', [&>'] = a>/3 + &//3',
where a, a', ft, @' denote matrices whose elements are integers.
* And not zero, since nlWl + ... + npWp cannot be a constant. Cf. for instance, Neumann,
Riemann's Theorie der AbeVschen Integrate (Leipzig, 1884), p. 247, or Forsyth, Theory of
Functions (1893), p. 447. (Riemann, Werke, 1876, p. 124.)
t As previously remarked, p. 247, note.
J A circuit of (&,•) in the positive direction furnishing a contribution of - 1 to a,-(,.
34—2
532 TRANSFORMATION OF PERIOD LOOPS. [322
If L1} ..., Lp be a set of p integrals of the second kind associated with
ul} ..., up, as in § 138, Chap. VII., and satisfying, therefore, the condition
x, a-. r> / \ a2 t\ \ / \ ®X
i ] = Dx \(z, x) T \- Dz \(x, z) -r ,
L atj L acJ
and the period matrices of Llt ..., Lp at the original and general period
loops be denoted respectively by - 2rj, - 2?/ and - [2iy], - [2^'], we have,
similarly, for the same values of a, a', /3, /3',
[?;] = 7/-« + 7?'a', [r/] = ??/3 + T//3'.
We have used the notation flp for the row of P quantities 2o>P + 2&/P',
where P, P' each denotes a row of p quantities ; we extend this notation to
the matrix 2eoa + 2a/a', where a, a' each denotes a matrix of p rows and
columns, and denote this matrix by Qa ; similarly we denote the matrix
2r)d + 2?/a' by Ha ; then the four equations just obtained may be written
[2»] = n.f [2a/] = fV [27,] = #a, [27/] = #0. (I.)
Noticing now that the matrices [2<o], [2o>'], [2?;], [2?/] must satisfy the
relations obtained in § 140, we have
iTrt = [ifl [»'] - [5] [V] =
= a
= (a/3' - a'/3) ^wt,
in virtue of the relations satisfied by the matrices 2<o, 2a/, 2i;, 2?/; and
similarly
0 = [fj] [»] - [5] [T/] = i (^aOa - HaJffa) = (oV - a'a) J^ri,
and
0 = [5H M - [«1 M = 1 (Hefy ~ n^) = (^/3' - /S'/S) i^ri ;
thus we have
off -30 = 1 =&*-&*', aa'-a'a = 0, /3/S' - /S'/3 = 0, (II.)
namely, the matrices a, /S, of, /3' satisfy relations precisely similar to those
respectively satisfied by the matrices to, co', 77, 77 x, the ^-m' which occurs
for the latter case being, in the case of the matrices a, /3, a', /3', replaced
by — 1 ; therefore also, as in § 141, the relations satisfied by a, /3, a', /S' can be
given in the form
a/S'-/Sa/ = l=/9/a-a//S, a/S-/Sa=0, a'/S' - /SV = 0. (III.)
In virtue of these equations, if
denote the matrix of 2p rows and columns formed with the elements of the matrices a, £,
a', £', we have (cf., for notation, Appendix ii.)
a,/3W ^', -j8\ /a|8'-/3a', /3a-aj8\/10\
a', (87 V - 5', a^ \ftjf - jS'a', /3*d - *W \0 V '
323] TRANSFORMATION OF INTEGRALS. 533
and therefore
and the original periods can be expressed in terms of the general periods in the form
w = [w] /3' — [to'] a', a)' = — [w] £ + [to'] a,
If 0 denote the matrix of jo rows and columns whereof every element is zero, and
1 denote the matrix of p rows and columns whereof every element is zero except those in
the diagonal, which are all equal to 1, and if e denote the matrix of 2p rows and columns
given by
then it is immediately proved that the relations (II.), (III.) are respectively equivalent to
the two equations
where
•7" / o , Q
«/ = ( -' -
\P> P
and it will be noticed that the equations (III.) are obtained from the equations (II.) by
changing the elements of J into the corresponding elements of J.
It follows* from the equation JfJ=e that the determinant of the matrix J is equal to
+ 1 or to - 1. It will subsequently (§ 333) appear that the determinant is equal to +1.
Ex. Verify, for the case ^=2, that the matrices
_/ 4, -20\ fl/-29, 124\
, /-3, 20\ / 22, -124\
Q=V-8, -7> "=( 56, 43J
satisfy the conditions (III.) (Weber, CreUe, LXXIV. (1872), p. 72).
323. It is often convenient, simultaneously with the change of period
loops which has been described, to make a linear transformation of the
fundamental integrals of the first kind, u^, ...,up. Suppose that we intro
duce, in place of ult ..., up, other p integrals w1} ..., wp, such that
ui = Miilwl+ +Mi>pwp,
or, as we shall write it, u = Mw, M being a matrix whose elements are
constants and of which the determinant is not zero. We enquire then what
are the integrals of the second kind associated with w1} ..., wp. We have
(§ 138) denoted Dui by /*;(#), and the matrix of the quantities /*;(<?;) by p, ;
* For another proof of the relations (II.), (HI.) of the text, the reader may compare Thomae,
Crelle, LXXV. (1873), p. 224. A proof directly on the lines followed here may of course be
constructed with the employment only of Riemann's normal elementary integrals of the first
and second kind. Cf. § 142.
534 TRANSFORMATION OF PERIODS AND [323
denote now, also, Dwf a by pt (x), and the matrix of the quantities pt (GJ) by p ;
then we immediately find //. = pM, and the equation (§ 138)
Lx> a = p-lHx> a - 2aux> a
gives
ML*' a = p~lHx' a - 2MaMwx> a ;
thus the integrals of the second kind associated with w1} ...,wp are the p
integrals given by MLx>a, and, corresponding to the matrix a for the
integrals L^ , ..., Lp' , we have, for the integrals MLx'a, the matrix
a = MaM. If 2u, 2i/ denote the matrices of the periods of the integrals w,
and — 2£, — 2£" denote the matrices of the periods of the integrals MLx-a, so
that (§ 139)
we therefore have w = Mv, w = Mv and
£ = 2MaMv = Mi), Z' = 2MaMv'-±MfJL-^ = M'r)'; (IV.)
it is immediately apparent from these equations that the matrices v, v, £ £"
satisfy the equations of § 140,
vv' - v'v = 0, £? - £' ? = 0, v'l -vl' = \ Tri = &' - ?v.
324. The preceding Articles have sufficiently shewn how the equations
of transformation of the periods arise by the consideration of the Abelian
integrals. It is of importance to see that equations of the same character,
but of more general significance, arise in connexion with the analytical
theory of the theta functions.
Let <w, &>', 77, ?;' be any four matrices of p rows and columns satisfying
the conditions (i) that the determinant of &> does not vanish, (ii) that co~la)'
is a symmetrical matrix, (iii) that the quadratic form co^tw'w2 has its
imaginary part positive when nly ...,np are real, (iv) that r)a)~l is a sym
metrical matrix, (v) that rj = t]w~lo>' — ^Triw~l. The conditions (i), (ii), (iv),
(v) are equivalent to equations of the form of (B) and (C), § 140, and,
taking matrices a, b, h such that a = ^t}(o~1) h = ^7riw~l, b = Triw^co', or
2hw = TTI, Zha)' = b, i] = 2a&), ?/ = 2aa>' — h, the condition (iii) ensures the
existence of the function defined by
^. (u . <2'\ _ ^eo«2+2to(n+Q')+6(n+Q')2+2«Q(n+Q'))
wherein Q, Q' are any constants (cf. § 174).
Introduce now two other matrices [&>], [&/], also of p rows and columns,
defined by the equations
[&>] = cow + &)V, = ^Ha, say, [CD'] = (u/3 + &//3', = £00, say,
where a, a', /3, ft', are matrices of p rows and columns whose elements are
824] ARGUMENTS OF A THETA FUNCTION. 535
integers*, it being supposed i" that the determinant of the matrix [<w] does
not vanish ; and introduce p other variables wi} ..., wp denned by
m= Mif !«/! + ...... +Miipwp, (i= 1, 2, ...,p)
or u = Mw, where M is a matrix of constants, whose determinant does not
vanish; let the simultaneous increments of wl} ...,wp when ult ...,up are
simultaneously increased by the constituents of the j-th column of [<w] be
denoted by vltj, . .., vpj, and the simultaneous increments of wl> . .., wp
when MJ, ...,up are simultaneously increased by the elements of the j-ih
column of [«'] be denoted by v'^j, ...,V'PIJ\ then we have the equations
2Mv = 2 [CD] = fta, 2Mi/ = 2 [&>'] = Hp, where u, i/ denote the matrices of
which respectively the (i,j) elements are Vij and v'ij.
The function S-(w; ^) is a function of w1; ..., wp; we proceed now to
investigate whether it is possible to choose the matrices a, a.', /3, ft' and the
matrix M, so that the function may be regarded as a theta function in
wlt ...,wp of order r (cf. Chap. XV. § 284).
Let the arguments wl} ..., wp be simultaneously increased by the con
stituents of the j-th column of the matrix 2t>; thereby u^, . .., up will be
increased by the constituents of the j-ih column of the matrix [2&>], and,
since a, a, /3, /3' consist of integers, the function S- (u ; ^ ) will (Chap. X.
§ 190) be multiplied by a factor e^i where
Z,- = (#.)«> [w + i(fla)tf>]-7rtt(«)(') (a')(j) +27rt'[(a)tf> Q'-(O(j) Q],
(a)(^ denoting the row of ^ elements forming the J-th column of the matrix
a, and (na)(-", (Ha}(h denoting, similarly, the j-ih columns of the matrices
2<»a + 2<a'a', 2?;a + 277 V respectively ; this expression Lj, is linear in w/u . . . , wp,
and can be put into the form
where (wlt ..., wp) denotes the row letter whose elements are wlt ..., wp, and
similarly (vltj, ..., vp>j) is the row letter formed by the elements of the j-th
column of the matrix v, r is a positive integer which is provisionally
arbitrary, Kj and 2£1|j, ..., 2%pj are properly chosen constants, and
(%£ij, ..., 2^i?-) is the row letter formed of the last of these. Similarly, if
the arguments w1} ...,wp be simultaneously increased by Zv'^j, ... , 2v'pj, the
function S- (u ; ^ ) takes a factor e^'i, where
Lf = (Hf)<* [u + ± (Q,)(J>] - m (/3)0') (/3')U> + 2^ [(ft)* Q - (P)& Q],
and, with the same value of r, this can be put into the form
* The case when a, a', j8, /3' are not integers is briefly considered in chapter XX.
t We have jn'ur1 [w] = iria + ba' ; we suppose that the determinant of via + ba' does not
vanish.
536 TRANSFORMATION OF PERIODS AND [324
where Kj, £"1)7-, ..., £'pj are properly chosen constants. In these equations
we suppose^ to be taken in turn equal to 1, 2, . .., p.
Comparing the two forms of Lj we have
or
so that the (i, j)th element of the matrix MHa is 2r£ij; hence if f, £' denote
respectively the matrices of the quantities ^j and %'ij, we have
toa = 2r£ MH? = 2< ; (V.)
from these we deduce, in virtue of the equations *2Mv = £la, 2Mv = Zip,
and therefore, in particular, comparing the (j, j)ih elements on the two sides
of these equations,
where, as before, (t/)^ is the row letter formed by the elements of the j-th
column of the matrix v, etc.; therefore the only remaining conditions
necessary for the identification of the two forms of Lj and L/, are
Kf = (a)0 Q' - (a')"' Q - 1 (a)U> (a')'*, - Kj = (/3)(j) Q' - (/3')(j) Q -
and the j9 pairs of equations of this form are included in the two
K' = «Q' -a'Q-^d (fia'), -K = 0Q- 0'Q - $d (0/3'), (VI)
where K', K are row letters of p elements and d (aaf), d(ft/3') are respectively
the row letters of p elements constituted by the diagonal elements of the
matrices aa', y9/3'.
The equations (VI.) arise by identifying the two forms of Lj and Z/; it is
effectively sufficient to identify the two forms of eLJ and eLi'\ thus it is
sufficient to regard the equations (VI.) as congruences, to the modulus 1.
We now impose upon the matrices v, v', £, £' the conditions
f«, - X= 0 = I 'v' - V?, lv - V? = %7Ti, (VII.)
which, as will be proved immediately, are equivalent to certain conditions
for the matrices a, /3, 'a.', /3'; then, denoting *r(u; J) by <f>(wly ...,wp) or
<f>(w), it can be verified* that the 2p equations
<t>(...,Wr + 2vrJ,...) = eL><l>(w\ $(..., wr + 2v\.>i,...) = eL>'<t>(w),(j = l,...,p\
where Lj, L/ have the specified forms, lead to the equation
<j> (w + %vm -f 2l»'m') = er^m+2£ m'> (w+vm+v'm1)— rmmm'+2Tri(mK'-m'K) A /^,\
wherein m, m' are row letters consisting of any p integers ; and this is the
* The verification is included in a more general piece of work which occurs in Chap. XIX.
324] ARGUMENTS OF A THETA FUNCTION. 537
characteristic equation for a theta function of order r with the associated
constants 2u, 2i/, 2£ 2£' (§ 284, p. 448).
The equations (VII.) are equivalent to conditions for the matrices v, v\
£, £', entirely analogous to the conditions (ii), (iv), (v) of § 324 for the
matrices w, &>', 77, 77'. The condition analogous to (i) of § 324, namely that the
determinant of the matrix v do not vanish, is involved in the hypothesis
that the determinant of Trio. + ba.' do not vanish. It will be proved below
(§ 325) that the remaining condition involved in the definition of a theta
function, viz. that the quadratic form v~lv'n* has its imaginary part positive
for real values of nlt ..., np> is a consequence of the corresponding condition
for the matrices &>, &>'. We consider first the conditions for the equations
(VII).
In virtue of equations (V.), the equations (VII.) require
= 4r ($/ - v?) =
and, similarly,
HaCla - iiJJ. = 0, Hftflft - npHft = 0 ;
but
') (&>/3 + o'/S') - (aa> + oV) (77/3 + T//3'),
j3' + a (rf(*> - 0/77) /3 + a' (tfw - wrf} $',
and this, by the equations (B), § 140, is equal to
i7n'(a/3'-a'/3);
thus
a/3' - a'/3 = yg'a - yga' = r, ( VIII.)
and, similarly,
aa'-o'a = 0, /9/g' - J3'/3 = 0 ;
and as before (§ 322) these three equations can be replaced by the three
ay8 = /Sa, of ft' = /Pa', *$' - $*' = r = jS'a- *'$, (IX.)
the relations satisfied by the matrices a, /9, a', £' respectively being similar to
those satisfied by o>, o>', 77, 77', with the change of the ^-rri, which occurs in the
latter case, into — r.
The number r which occurs in these equations is called the order of the
transformation; when it is equal to 1 the transformation is called a linear
transformation.
Ex. i. Prove that, with matrices of 2p rows and 2p columns,
« 0W ? -0\ A 0\ /« a'
ana
/a 0WO -IWa a'\ /O - 1\
W/r/v oA/3)3';-rVi o;-
The determinant of the matrix will be subsequently proved to be
538 CONVERGENCE OF TRANSFORMED FUNCTION. [324
Ex. ii. Prove that the equations (V.) of § 324 are equivalent to
(M 0 \ /2v 2iA _ /2o> 2«A fa |3 \
Vo RfftV W «V " k*i *i') V ft'J '
Ex. iii. If x, y, xlt y± be any row letters of p elements, and X, Y, Jfo Y± be other
such row letters, such that
f . X
tey), ° T
then the equations (VIII.) are the conditions for the self-transformation of the bilinear
form xy± — x$, which is expressed by the equation
XTl-XlY^r(xy^-xly).
325. Conversely when the matrices a, a, j3, ft satisfy the equations
(VIII.), the function ^ (u ; ^ ) satisfies the determining equation for a theta
function in w1} ..., wp, of order r, with the characteristic (K, K'}, and with
the associated constants 2u, 2t/, 2f, 2£"; and in virtue of the equations (VII.),
the determinant of v not vanishing, matrices a, b, h, of which the first two
are symmetrical, can be taken such that
a = \ £v~l, h = ^ Triv~l, b = 7riv~l v' ;
we proceed now to shew* that the real part of the quadratic form bw3 is
negative for real values of nl} ...,np> r being positive, as was supposed.
The quantity, or matrix, obtainable from any complex quantity, or
matrix of complex quantities, by changing the sign of the imaginary part
of that quantity, or of the imaginary parts of every constituent of that
matrix, will be denoted by the suffix 0 ; and a similar notation will be used
for row letters ; further the symmetrical matrices w~l(a, v~lv will be denoted
respectively by T and r', so that b = TTIT, b = TTIT' ; also r, r will be written,
respectively, in the forms TL + irz> T/ + ira', where T!, r2, iV, r2' are matrices
of real quantities. Then, putting
x = vM(a~[x, and therefore #„' = V^MQW^XQ,
where x, x denote rows of p complex quantities, and x0', x0 the rows of the
corresponding conjugate complex quantities, and recalling that
r'^r' = vv-1, a>-1Mv = OL + TCL', a)~lM v = /3 + rft,
we have _ _ _
^x . V^M^W^XQ = v'MwT^x . v^M^^Xo
and, if x = xl + ix2, sc0 = x1 — ix^, where xlf x2 are real, this is equal to
(]3 + ft^ + ift-r.?) (^ + ix2) . (a + S'T!
or
\ftP + ftP' + i (J3Q + ftQ'}} [5P + fi'P' - i (aQ + a'Q')],
* Hermite, Compt. Eendus, XL. (1855), Weber, Ann. d. Mat., Ser. 2, t. ix. (1878—9).
326] SIMPLEST FORM OF THE LINEAR TRANSFORMATION. 539
where P, P', Q, Q' are row letters of p real quantities given by
P = Xlt P = T^ - T2#2, Q = X2, Q'= 7V
so that
thus the coefficient of i in T'X'XQ is
(fiP + a'F) 08Q + ft'Q') - (ftP + ft'P') («Q + a'Q'),
which, in virtue of the equations (IX.), is equal to r (PQ' — P'Q) or
rrz (x? + #22) ; thus the coefficient of i in r'afx^ is equal to the coefficient
of i in rTXxn. Since x may be regarded as arbitrarily assigned this proves
that the imaginary part of r'x'x0 is necessarily positive ; and this includes
the proposition we desired to establish.
Ex. Prove that the equation obtained is equivalent to
J/Q VQ T2'vM = rWQ T2 <5.
326. Of the general formulae thus obtained for the transformation of
theta functions, the case of a linear transformation, for which r=l, is of
great importance ; and we limit ourselves mainly to that case in the
following parts of this chapter. We have shewn that a theta function of the
first order, with assigned characteristic and associated constants, is unique,
save for a factor independent of the argument ; we have therefore, for r = 1,
as a result of the theory here given, the equation
We suppose a, ft, a, ft' to be any arbitrarily assigned matrices of integers
satisfying the equations (VIII.) or (IX.); then there remains a certain
redundancy of disposable quantities ; we may for instance suppose co, w , 77, T/
and M to be given, and choose v, v, £, £' in accordance with these equations ;
or we may suppose to, &>', v, % and £' to be prescribed and use these equations
to determine M, v', r) and ?/. It is convenient to specify the results in two
cases. We replace u, w respectively by U, W.
(i) 2o) = 1, 2co' = r , T; = a, T/' = ar — m, h = TTI, b = TTIT,
2u = 1, 2i/ = T', £ = 0, £" = — iri , a = 0 , h = iri , b = TTI'T',
U = MW, M = a + ra', (a + TO!) r' = ft + rft',
so that, as immediately follows from equations (IX.),
Ta') = r = (/3/-aV)(a + aT), U=(a + Ta')W, W =\(& -r'a.')U,
and, because ij' = rjr — TTI and £ = 0,
|
from which we get
7T"? —
a = 77 = iria! (a + ra')'1 = ~ a' (ft' - r'a),
540 SIMPLEST FORM OF THE LINEAR TRANSFORMATION. [326
a U* = — «' (£' - T'O') tf2 = TriV F U = Trt V (a + ra') Fa.
These equations satisfy the necessary conditions, and lead, when r = 1, to
'; *') f (X.)
where A is independent of £/!, ..., Up, and the characteristic (K, K'} is deter
mined from (Q, Q'} by the equations (§ 324)
IT = aQ' - a'Q - id (aa'), -K = 0Q[
The appearance of the exponential factor outside the e-function, in equation (X.),
would of itself be sufficient reason for using, as we have done, the 5-function, in place of
the 6-function, in all general algebraic investigations*.
If in § 324 we put
we easily find
via1 (a + ra') TF2 = ^o> ~ %2 - irfw ~ %2 ;
thus (§ 189, p. 283) equation (X.) includes the initial equation of this Article.
In general the function occurring on the left side of equation (X.) is
a theta function in W of order r with associated constants 2v — 1, 2i/ = T',
2£=0, 2£'=-27rt, and characteristic (K, K').
(ii) A particular case of (i), when the matrix a! consists of zeros, is given
by the formulae
2&) = 1, 2&>' = T , V) = 0, t\ = — TTI, a = 0, A = 7n, 6 = TH'T ,
2u = 1, 2i/ = T', £ = 0, £' = - Tri, a = 0, h = TTt, b = TUT',
£7 = « If, T = a-1 (£ + r/3'), r = - (err7 - /3) a,
r
/a £ \ /a /3 \ , 5
' /o' = A — i > wnere a£ = /3a.
\a p 1 \0 ra V
Then the function @((7; r; «') or © [a W; 1 (ar7 - /9) a ; |] is a theta
function in W, of order r, with associated constants 2u = 1, 2i»' = r/, 2^=0,
2 ^" = — 2-Tn, and characteristic (.ST, K') given by
and, in particular, when r = 1 we have
.'-- : , '' e(tr;r;f)=4@(Tf;T';5')) " (XL)
where A is independent of U1} ..., Z7P.
* Of. § 189 (Chap. X.); and for the case _p = l, Cayley, Liouville, x. (1845), or Collected
Works, Vol. i., p. 156 (1889).
327] TRANSFORMATION OF CHARACTERISTICS. 541
327. It is clear that the results just obtained, for the linear trans
formation of theta functions, contain the answer to the enquiry as to the
changes in the Riemann theta functions which arise in virtue of a change in
the fundamental system of period loops. Before considering the results in
further detail, it is desirable to be in possession of certain results as to the
transformation of the characteristics of the theta function, which we now
give ; the reader who desires may omit the demonstrations, noticing only the
results, and proceed at once to § 332. We retain the general value r for the
order of the transformation, though the applications of greatest importance
are those for which r = 1.
As before let ^(7) denote the row of p quantities constituted by the
diagonal elements of any matrix 7 of p rows and columns ; in all cases here
arising y is a symmetrical matrix ; then we have
a d 03/3') + /3d (aa') = rd(aft ), ft'd (aft) + fid (aft') = rd (ft ft')
_ _ _ (mod. 2)
ad (ft ft') + ft'd (aa') = rd (a' ft'), *'d (aft) + ad (a/3') = rd(aa')
and
d (a*') d (#3') = (r+l) 2d (fta) = (r + l) 2d (ft'a)
- - _ (mod. 2),
d (a/9) d (a' ft') = (r + I) 2d (aft') = (r + 1) 2d (ft a')
so that, when r = 1 or is any odd integer,
d (aa) . d (ft/3') = d (aft) . d (a' ft') = 0 (mod. 2).
The last result contains the statement that the linear transformation of
the zero theta-characteristic is always an even characteristic.
For the equations
ft'd -a' ft = r, aft = ftd,
give
aft ft'a -ftaa' ft = raft,
and therefore
ftft'z*-aay = raftx2,
where x is any row letter of p integers, and z—ax, y = $x; but if y be a symmetrical
matrix of integers and t be any row letter of p integers yp, =yn^2 + ... + 2y12^2-f..., is
= yn'i2+ ••• + Wp2> and therefore =yut1+... +ypptp, or =d(y). t, for modulus 2 ; hence
d (ftp) z-d (aa') y = rd (aft) x (mod. 2)
or
[ad (ftp) +pd (aa') - rd (aft)] x = 0 (mod. 2) ;
and as this is true for any row letter of integers, #, the first of the given equations follows
at once. The second of the equations also follows from /3'a - a'J3 = r, in the same way, and
the third and fourth follow similarly from pa — fia=r.
To prove the fifth equation, we have, since PS — aft = r,
ftft'aa' = fta'fta'
or
542 PROPERTIES UNALTERED BY LINEAR TRANSFORMATION. [327
where b=ftftr, a = aa, c = fta' ; hence, equating the sums of the diagonal elements on the two
sides of the equation, we have
p P p p P
2T 7). -n • • — ^ T P' ./>. .-L.V *9 s> . .
£i k% j MJ, i — ^ ^ ^j V C-* j T- / 2i Cj • ^ ,
j=l i=l j=l i=l i=l
therefore, as, unless i=j, bitjO,jti = bj<iai>j, because a, b are symmetrical matrices, and as
we obtain
p P P
2 aiti\i= 2 (c2i>i+rc1-)i) = (r+ 1) 2 c,-jt-.
The sixth equation is obtained in a similar way, starting from ft'a — fta' = r.
Of the results thus derived we make, now, application to the case when r is odd, limiting
ourselves to the case when the characteristic (Q, Q') consists of half-integers ; we put then
Q — fyi Q' = Wi so that q, q' each consist of p integers ; then K, K' are also half-integers,
respectively equal to \k, \k' , say, where
k' = aq' -a'q-d (aa'), -k = ftq'-ft'q-d (ft ft').
In most cases of these formulae, it is convenient to regard them as congruences, to
modulus 2. This is equivalent to neglecting additive integral characteristics.
From these equations we derive immediately, in virtue of the equations of the present
Article
q=ak + ftk' + d(aft), q' = a'k + ft'k' + d (a' ft') (mod. 2)
and
qq' = kk' (mod. 2).
Further if p, p' be row letters of p integers, and
v' = afj.' — a'p. — d(aa), —v=ftp — ft'fj. — d (ftff),
we find, also in virtue of the equations of the present Article,
kv - k'v = qp - q'fi. + (p + q') d (aft) + (n + q)d (aft'), (mod. 2) ;
therefore, if also
o-' = ap - a'p - d (aa), —a~ = ftp' — ft'p — d (ft ft'),
we have
k v - k'v + vcr — v'cr -f o-fc' — a'k = qp — q'p. + pp — p'p + p q' — p'q (mod. 2 ).
Denoting the half-integer characteristics i ( ^ ) . A y } . A ( p } by A, B, C,
2 V? / 2 W VP / *
,7/, . A / '\
and the characteristics A(7 1, Jr'f 1, |f ( J, which we call the transformed
\K I \v J \cr J
characteristics, by A', B', C', we have therefore the results (§ 294)
A\= A'\, \A,B,C =\A',B',C'\, (mod. 2)
or, in words, in a linear transformation of a theta function with half-integer
characteristic, and in any transformation of odd order, an odd (or even)
characteristic transforms into an odd (or even) characteristic, and three
syzygetic (or azygetic) characteristics transform into three syzygetic (or
azygetic) characteristics.
Of these the first result is immediately obvious when r = \ from the equation of
transformation (§ 326), by changing w into —w.
THETA CHARACTERISTICS AND PERIOD CHARACTERISTICS. 543
Hence also it is obvious that if A be an even characteristic for which
; A) vanishes, then the transformed characteristic A' is also an even
characteristic for which the transformed function ^ (0 ; A') vanishes.
328. If in the formula of linear transformation of theta functions with
half-integer characteristic, which we may write
we replace u by u + ^£lm = u + com + a>'m' , where m, m' denote rows of
integers, and, therefore, since w =M(vJ3' — v'af), w — M (— v/3 + i/a), (cf. Ex. i.,
§ 324), replace w by w + vn + v'n', where
n' = am' — a'm, — n = $m! — J3'm,
we obtain (§ 189, formula (L))
f
\u ;
|_ '
= <«• a \w ;
-
where A' is independent of u1} ..., up, and k' + n', k + n are obtainable from
q' + m', q + m by the same formulae whereby k', k are obtained from q', q,
namely
k' + m' = a (q' + m') -a'(q + m)-d (aa'),
- (k + m)=0(q' + mf) - ff (q + m)-d(J3ff);
these formulae are different from those whereby n', n are obtained from
m', m ; for this reason it is sometimes convenient to speak of 1 f ^ ] as a theta
2\qJ
characteristic, and of ^ I ) as a period characteristic ; as it arises here the
difference lies in the formulae of transformation ; but other differences will
appear subsequently; these differences are mainly consequences of the
obvious fact that, when half-integer characteristics which differ by integer
characteristics are regarded as identical, the sum of any odd number of
theta characteristics is transformed as a theta characteristic, while the
sum of any even number of theta characteristics is transformed as a
period characteristic. In other words, a period characteristic is to be
regarded as the (sum or) difference of two theta characteristics.
It will appear for instance that the characteristics associated in §§ 244, 245,
Chap. XIII. with radical functions of the form JX (2"+1> are to be regarded as
theta characteristics — and the characteristics associated in § 245 with radical
functions of the form JX^, which are denned as sums of characteristics
associated with functions JX(*"+v, are to be regarded as period characteristics.
544 LINEAR TRANSFORMATION OF ANY EVEN [328
We may regard the distinction* thus explained somewhat differently, by taking as the
fundamental formula of linear transformation that which expresses # \ u ; £ ( j in terms
where
Q,.; |ffj L
and
l' = kr + d(aa') = aq'-aq, -1= - k + d ($$} = &q' - ffq.
In the following pages we shall always understand by ' characteristic,' a
theta characteristic ; when it is necessary to call attention to the fact that a
characteristic is a period characteristic this will be done.
329. It is clear that the formula of linear transformation of a theta
function with any half-integer characteristic is obtainable from the particular
case
where r' = d (act), r = d (ft ft), by the addition of half periods to the argu
ments. It is therefore of interest to shew that matrices a, ft, of, ft' can be
chosen, satisfying the equations
aft = ft a, a' ft' = ft' a, aft' -fta=l,
ir'\
which will make the characteristic ^ I j equal to any even half-integer
characteristic.
Any even half-integer characteristic, being denoted by
/If ' If '\
1 Ki •••Kp \
2 I J, I. >
\n,i . . . Up J
lk'\
we may, momentarily, call ( * ) the i-th column of the characteristic ; then
V^i /
the columns may be of four sorts,
o'
but the number of columns of the last sort must be even ; we build now a
matrix
* *}
' ft')
* Theta characteristics have also been named eigentliche Charakteristiken and Primcharak-
teristiken ; they consist of 2>'-1(2P-l) odd and 2P~1(2*> + 1) even characteristics. The period
characteristics have been called Gruppencharakteristiken and Elementarcharakteristiken or
sometimes relative Charakteristiken. For them the distinction of odd and even is unimportant —
while the distinction between the zero characteristic — which cannot be written as the sum of two
different theta characteristics — and the remaining 22P - 1 characteristics, is of great importance.
The distinction between theta characteristics and period characteristics has been insisted
on by Noether, in connection with the theory of radical forms — Cf. Noether, Math. Annal.
xxvin. (1887), p. 373, Klein, Hath. Annal. xxxvi. (1890), p. 36, Schottky, Crelle, en. (1888),
p. 308. The distinction is in fact observed in the Abel'sche Functionen of Clebsch and Gordaii,
in the manner indicated in the text,
329]
CHARACTERISTIC INTO THE ZERO CHARACTERISTIC.
545
of *2p rows and columns by the following rule* — Corresponding to a column
of the characteristic of the first sort, say the i-th column, we take &i,i=l3>i,i= 1 ,
but take every other element of the t'-th row and t'-th column of a and /3',
and every element of the i-th row and i-th column of $ and a' to be zero ;
corresponding to a column of the characteristic of the second sort, say the
j-ih column, we take «/,_,- = P'jj = afjj — 1, but take every other element of
the j-th row and j-th column of a, /?', a', and every element of the j-th row
and column of @, to be zero ; corresponding to a column of the characteristic
of the third sort, say the ra-th column, we take am>m = fim,m = ftm,m = 1, but
take every other element of the m-th row and column of a, /3, /3' and every
element of the m-th row and column of «' to be zero ; corresponding to a pair
of columns of the characteristic of the fourth sort, say the p-ih and cr-th, we
take ap> p = /3P) p = j3'pi p = l, a<r>tr = a^ „ = ft'*, « = 1, a^ p = 1, /8pt v = - 1, a!Vt p = 1,
/3'p)(r = — 1, and take every other element of the p-ih row and column and of
the o--th row and column, of each of the four matrices a, a', yS, /3', to be zero.
Then it can be shewn that the matrix thus obtained satisfies all the
necessary conditions and gives k' = d (««'), k = d (/S/3').
Consider for instance the case p = 5, and the characteristic
/O 1 0 1 1\
\0 0 1 1 \)
the matrix formed by the rules from this characteristic is
1
0
0
o
0
0
0
0
0
0
0
1
0
0
0
'o
0
0
0
0
0
0
1
a
0
0
0
1
0
0
0
0
0
i
0
0
0
0
1
_ J
0
0
0
i
1
0
0
0
0
0
0
0
0
o
0
1
0
0
0
0
0
1
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
1
_ J
0
0
0
1
1
0
0
0
0
1
and it is immediately verified that this satisfies the equations for a linear transformation
(§ 324 (IX.), for r = l), and gives, for the diagonal elements of aa', J3f¥, respectively, the
elements 01011 and 00111.
Since we can transform the zero characteristic into any even characteristic, we can of
course transform any even characteristic into the zero characteristic ; for instance, when
there is an even theta function which vanishes for zero values of the arguments, we can
by making a linear transformation, take for this function the theta function with zero
characteristic.
* Clebsch and Gordan, Abel. Fctnen (Leipzig, 1HGO), p. 318.
35
546
TRANSFORMATION OF ANY AZYGETIC SYSTEM
[329
Ex. For the hyperelliptic case, when jo = 3, the period loops being taken as in § 200,
the theta-function whose characteristic is \ ( , ., , ) vanishes for zero arguments (§ 203) ;
prove that the transformation given by
a = ( 100), /3=(
-1 00), a' = ( 00-1
), £' = (
101)
010
0-10
loo o
010
-101
1 00
00-1
001
is a linear transformation and gives an equation of the form
where A is independent of ul9 ..., up.
330. We have proved (§327) that if three half-integer theta character
istics be syzygetic (or azygetic) the characteristics arising from them by any
linear transformation are also syzygetic (or azygetic). It follows therefore
that a Gb'pel system of 2r characteristics, syzygetic in threes (§ 297, Chap.
XVII.), transforms into such a Gbpel system. Also the 22<r Gb'pel systems of
§ 298, having a definite character, that of being all odd or all even, transform
into systems having the same character. And the 2o- + 1 fundamental Gopel
systems (§ 300), which satisfy the condition that any three characteristics
chosen from different systems of these are azygetic, transform into such
systems ; moreover since the linear transformation of a characteristic which
is the sum of an odd number of other characteristics is the sum of the
transformations of these characteristics, the transformations of these 2cr + 1
systems possess the property belonging to the original systems, that all the
22<r Gopel systems having a definite character are representable by the
combinations of an odd number of them. It follows therefore that the
theta relations obtained in Chap. XVII., based on the properties of the
Gb'pel systems, persist after any linear transformation.
331. But questions are then immediately suggested, such as these : What are the
simplest Gopel systems from which all others are obtainable* by linear transformation1?
Is it possible to derive the 22<r Gopel systems of § 298, having a definite character, by
linear transformation, from systems based upon the 22<r characteristics obtainable by taking
all possible half-integer characteristics in which p — a- columns consist of zeros ? Are the
fundamental sets of 2p + 1 three-wise azygetic characteristics, by the odd combinations of
which all the 22p half-integer characteristics can be represented (§ 300), all derivable by
linear transformation from one such set ?
We deal here only with the answer to the last question — and prove the following
result: Let D, Dlt ..., AP + I be any 2p + 2 half -integer characteristics, such that, for i<j,
* An obvious Gopel group of 2v characteristics is formed by all the characteristics in which
the upper row of elements are all zeros, and the lower row of elements each = 0 or £ .
331] INTO A STANDARD AZYGETIC SYSTEM. 547
i=l, ..., 2p, j = 2, ..., 2/>-fl, we have \D, Dt, D, =1 ; then it is possible to choose a half-
integer characteristic E, and a linear transformation, such that the characteristics
ED, EDlt ..
transform into
0, Xu ...,
where Xj, ..., X2P + 1 are certain characteristics to be specified, of which (by § 327) every two are
azygetic. It will follow that if D', D^, . . . , D'2P + 1 be any other set of 2p + 2 characteristics
of which every three are azygetic, a characteristic E', and a linear transformation, can be
found such that, with a proper characteristic E, the set ED, ED±, ..., ED2p + l transforms
into E'D1, E'Di, ..., E'D'^p + l. It will be shewn that the characteristics X1} ...,X2p + 1
can be written down by means of the hyperelliptic half-periods denoted (§ 200) by ua> c\
ua,a^ ua,c^ ^ ua'a", ua'° ; it has already been remarked (§ 294, Ex.) that the charac
teristics associated with these half-periods are azygetic in pairs. The proof which is to be
given establishes an interesting connexion between the conditions for a linear transforma
tion and the investigation of § 300, Chap. XVII.
Taking an Abelian matrix,
/- P-\
V/37'
for which
aa'-a'a = 0, jS/S' - J3'/3 = 0, ap -&'&=!,
define characteristics of integers by means of the equations
where a'g>r is the r-th element of the s-th row of the matrix a', etc. and r=l,2, ... , p • tfeen
the symbol which, in accordance with the notation of § 294, Chap. XVII., we define by the
equation
Mr> •5»i=ai,r£'i>*+'"+ap,rP>'p,«~a/l,>-P>l,«~ ••• — O-'p, r Pp,s>
is the (?•, s)-th element of the matrix a/rf-a'p1, and may be denoted by (a^'-a'/3)r)g; thus
the conditions for the matrices a, a', /3, p" are equivalent to the p (2p—l) equations
\Ar,Br\ = l, \Ar,B8\=0, 1^,^1=0, \J3r,Bs=0, (r=j=«, r, s = l, 2, ..., p),
whereof the first gives p conditions, the second p(p- 1) conditions, and the third and
fourth each \p(p— 1) conditions. It is convenient also to notice, what are corollaries
from these, the equations
\B,,Ar\=- Ar,£,=0, \Br,Ar\=-\Ar,Br\=-l, Br, Ar'\= -\Ar\Br\ = \Ar)Br\ = l.
Consider now the 2p + l characteristics, of integers, given by
whereof the first 2p are pairs of the type
for r=l, 2, ..., p, and a^b^ means the sum, without reduction, of the characteristics a/,
blt a.j, and so in general. The sum of these characteristics is a characteristic consisting
wholly of even integers. If these characteristics be denoted, in order, by cn c2, ..., c2p+l,
it immediately follows, from the fundamental equations connecting at, ..., 6P, that
35— -2
548
TRANSFORMATION OF AZYGETIC SYSTEMS.
[331
Thus the (2p + l) half-integer characteristics derivable from clt c2, ..., c-2p + 1, namely
Ci = i«i» — » CIP+I=&*P+I> are azJgetic in Pairs-
Conversely let Z>, D^ ..., D2p + l be any half-integer characteristics such that, for i<j,
i = l, ..., 2jt>, .7 = 2, ..., 2/> + l, we have | D, A, A =1» so that (§ 30°, P- 496) there exist
connecting them only two relations (i) that their sum is a characteristic of integers, and
(ii) a relation connecting an odd number of them ; putting Ci = DfDi(i=l, ..., *2p), where
2y= - D, we obtain a set of independent characteristics C1} ..., C2p, such that for i<j,
taking
tions
Thus putting ^ = 1^, ..., C'2J) +
the equations, previously given,
the i-th column of this matrix consisting of the elements of the lower and upper rows of
the integer characteristic af or bt, according as i<p + l or i>p. We proceed now to find
the result of applying the linear transformation, given by this Abelian matrix, to the
half-integer characteristics C1} ..., C'2P+1.
The equations for the transformation of the characteristic ( to the characteristic
j = 2, 3, ...,2p
where C"2)._1= — (72,._1, we have also the 2p equa-
= 1, (m=l, 2, ...,2p).
+ 1, we can obtain an Abelian matrix by means of
, which are (§ 324, VI.),
/?;/ = aj/-a/2'-c?(aa'), -k = $q' -ftq-d 0/3'),
are equivalent, in the notation here employed, to
# = 1^*, e|-[^(Sa')]i, -*i = |5t, § -[^(^')]i, (* = 1, 2, ...,
where ^=^4, ^^Jg-; taking
in turn, we immediately find that the transformations of the characteristics <72r-i>
1, are given, omitting integer characteristics, by
1...100...0\ ,/rf(fi«')\ ,/11...110...0\ ,/rf(aa')\ ,A1...
l...l 10.. .O' *d(pp) + *\II...IQO...O)' *VW)/ ^Vll-
or, say, by
rf(aa')\ ,
*
« , !
respectively.
Now let the characteristics
°\ f°
i/vp
be respectively denoted by
then we have proved that the half-integer characteristic DDi transforms, save for an
(r'\
), where r=c?(/3/3'), r' = d(aa); since the transforma-
331] NUMBER OF INCONGRUENT TRANSFORMATIONS. 549
tion of the sum of two characteristics is the sum of their transformations added to \ ( r\ ,
- -
and since the characteristic £( ), where s' = d(a'p'), s = d(ap), transforms into the zero
\s /
//\
characteristic (§ 327), it follows that the transformation of the characteristic |( )+DDi
\s /
is the characteristic Xj ; hence, putting E=$ ( } +D, and omitting integer characteristics,
v*/
the characteristics
ED, ED,, ..., ED2p + l
transform, respectively, into
0, Xj, ..., X2p + 1 ;
and this is the result we desired to prove.
The number of matrices of integers, of the form
in which aa-aa = 0, /3/3'-£'£=0, 5/3'-a'/3 = l, is infinite; but it follows from the
investigation just given that if all the elements of these matrices be replaced by their
smallest positive residues for modulus 2, the number of different matrices then arising is
finite, being equal to the number of sets of 2p + l half-integer characteristics, with integral
sum, of which every two characteristics are azygetic. As in § 300, Chap. XVII., this
number is
3 ...... (22-l)2;
we may call this the number of incongruent Abelian matrices, for modulus 2. Similarly
the number* of incongruent Abelian matrices for modulus n is
Ex. By adding suitable integers to the characteristics denoted by 1, 2, 3, 4, 5, 6, 7 in
the table of § 205, for p = 3, we obtain respectively
-100N /-i -ION /-l-lix ,/-101
-i o o)' o -i o' o o i' o i
00 -IN /O 1 -IN (01 ON.
1 o -I/1 *Vi i o/' *v i i/'
denoting these respectively by C^ C2, ..., Cr, we find, for i<j, that
The equations of the text
give
and therefore, in this case, we find
- 1 0 ON / - 1 0 IN /O 0 - 1
i n n > aZ = [ -, )i «<« = (
— 1 0 O/ \ — Illy \ 0 0 — 1
-1-1 ON ft _ / - 1 1 IN /Ol-l
* Another proof is given by Jordan, Traits des Substitutions (Paris, 1870), p. 176.
550
TRANSFORMATION OF AZYGETIC SYSTEMS.
[331
hence the ' linear substitution, of the text, for transforming the fundamental set of
characteristics Clt ..., C<j is
(-1-1 0 0-1 0)
1 0
1 -1
-1
0
1 -1 1 0 1-1
T7 *l.- C. A
from this we find
1
=f(1 a
\s / \(»(ap)/ \« ^
integral characteristic, it follows by the general theorem, that if the characteristic
matrix be applied, they will be transformed respectively into the characteristics X1? ..., \j.
A further result should be mentioned. On the hyperelliptic Riemann surface suppose
the period loops drawn as in the figure (12) ;
l 1 ' t n ,1 •
—\\ j, 3()=f(1 a )> since the sum of Clt ..., C7 is an
\(»(ap)/ \« ^ »/
llows by the general theorem, that if the characteristic
be added to each of 0^ ..., C7, and then the linear transformation given by the
FIG. 12.
then the characteristics associated with the half-periods ua> °l, ua> a\ ..., ua> Cp, ua> **",
ua' c will be, save for integer characteristics, respectively X1} X2, ..., X2P, X2P+1; this the
reader can immediately verify by means of the rule given at the bottom of page 297 of the
present volume.
Ex. Prove that if the characteristics 0, Xj,...., X2P + 1 be subjected to the transforma
tion given by the Abelian matrix of 2p rows and columns which is denoted by
then, save for integer characteristics, Xf is changed to Si + ^f J , where
1, -
0,
iN /ov
(0(o)
332] COMPOSITION OF TRANSFORMATIONS. 551
are the characteristics which arise in § 200, Chap. XI. as associated with the half-periods
ua>Cr, ua'ar, ua'° respectively. The characteristics 2n ..., 22p + 1 satisfy the p(2p-I)
conditions | 2(, 2/| = l, for i<j.
332. We proceed now to shew how any linear transformation may be
regarded as the result of certain very simple linear transformations performed
in succession. As a corollary from the investigation we shall be able to infer
that every linear transformation may be associated with a change in the
method of taking the period loops on a Riemann surface ; we have already
proved the converse result, that every change in the period loops is associated
with matrices, a, a', /3, /3', belonging to a linear substitution (§ 322).
It is convenient to give first the fundamental equations for a composition
of two transformations of any order. It has been shewn (§ 324) that the
equations for the transformation of a theta function of the first order, in the
arguments u, with characteristic (Q, Q') and associated constants 2eo, 2<u',
277, 27;', to a theta function of order r, in the arguments w, where u = Mw,
with characteristic (K, K'} and associated constants 2v, 2v, 2£, 2f, are
K' = aQ' - a'Q - \d (aa'), -K = $Q'- P'Q
M, 0 \ /2u, 2iA _ /2w, 2ftA /a ,
0, rM-*J fa 2(T/ ~ U?, 27/J U',
and from the last equation, writing it in the form /*U = OA, it follows, in
virtue of the equations OeH = - ^Trie, UeU' = — ^irie (§ 140, Chap. VII.), and
the easily verifiable equation JLep = re, where the matrix e is given by
O -
that also AeA = re, as in Ex. i., § 324. And, just as in § 324, it can be proved
that equations for the transformation of a theta function of order r in the
arguments w, with characteristic (K, K'), and associated constants 2u, 2i/, 2f,
2£', to a theta- function of order rs, in the arguments u1} given by w^Nu^,
with characteristic (Ql9 Q/), and associated constants 2o)!, 2ft)/, 27/j, 2?7/, are
Q/ = yK' - y'K - %rd (77 ), - & = 8K' - I'K -
N, _
0, «
and writing the last equation in the form vfli = UV, we infer as before that
^eV = se.
Now from the equations /u,U = HA, i/ni = UV, we obtain //-i>ni=;u,UV =
or, if A! = AV,
fMN, _0_ \ /2ft),, 2&>A = /2ft), 2ft)'\ .
V 0 , rtinSr-V UT;,, 2r/1/)~U^, 2V/
552 SUPPLEMENTARY TRANSFORMATIONS. [332
from this equation we find as before that the matrix A1} given by
A -AV-
Al - - '7 + /3V,
satisfies the equation Aj e Aj = rse. Similarly from the two sets of equations
transforming the characteristics, by making use of the equations
d (aja/) = 7<Z (oa') + yd (££') + rd (77),
d (&&') = $d (aa') + B'd (££') + rd (SS'), (mod. 2),
which can be proved by the methods of § 327, we immediately find
'), -Q,= M - fr'Q - \d (&&'), (mod-
Hence any transformation of order rs may be regarded as compounded of
two transformations, of which the first transforms a theta-function of the
first order into a theta function of the r-th order, and the second transforms
it further into a theta function of order rs.
It follows therefore that the most general transformation may be con
sidered as the result of successive transformations of prime order. It is
convenient to remember that the matrix of integers, A1? associated with
the compound transformation, is equal to AV, the matrix A, associated
with the transformation which is first carried out, being the left-hand
factor.
One important case should be referred to. The matrix
is easily seen to be that of a transformation of order r ; putting it in place of y, the final
equations for the compound transformation Vi may be taken to be
The transformation rA~: is called supplementary to A (cf. Chap. XVII., § 317, Ex. vii.).
333. Limiting ourselves now to the case of linear transformation, let
AK (k = 2, 3, . . . , p) denote the matrix of 2p rows and columns indicated by
^* = (A*t, 0 ),
0, /**
where fj,k has unities in the diagonal except in the first and &-th places, in
which there are zeros, and has elsewhere zeros, except in the &-th place of
the first row, and the k-ih place of the first column, where there are unities ;
let J9 denote the matrix of *2p rows and columns indicated by
333]
FOUR ELEMENTARY LINEAR TRANSFORMATIONS.
553
0
which has unities in the diagonal, except in the first and (p + l)-th places,
where there are zeros, and has elsewhere zeros except in the (p + l)-th place
of the first row, where there is — 1, and the (p + l)-th place of the first
column, where there is + 1 ; let C denote the matrix of 2p rows and columns
indicated by
which has unities everywhere in the diagonal and has elsewhere zeros,
except in the (p + l)-th place of the first row, where it has - 1 ; let D denote
the matrix of 2p rows and columns indicated by
D = (l 0-1 ),
1 -1 0
1 0
1 0
0 1
0 1
0 1
0 1
which has unities everywhere in the diagonal and has elsewhere zeros, except
in the (p + 2)-th place of the first row and the (p + l)-th place of the second
row, in each of which there is — 1. It is easy to see that each of these
matrices satisfies the conditions (IX.) of § 324, for r = 1.
Then it can be proved that every matrix of 2p rows and columns of
integers,
a ,
«',
554
ANY LINEAR TRANSFORMATION OBTAINABLE
[333
for which aft = fta, a' ft' = ft'a, aft' — @a? = 1, can be written* as a product of
positive integral powers of the (p + 2) matrices AZ) ..., Ap, B, C, D. The
proof of this statement is given in the Appendix (II) to this volume.
We shall therefore obtain a better understanding of the changes effected
by a linear transformation by considering these transformations in turn. We
have seen that any linear transformation may be considered as made up of
two processes, (i) the change of the fundamental system of periods, effected
by the equations
[<w] = coo. + co'a.', [&>'] = 6>/3 + aft',
[77] = 773 + 7/a', [77'] = 7} ft + rj'ft',
(ii) the change of the arguments, effected by the equation u = Mw, and
leading to
of these we consider here the first process. Applying the equations •}•
[CD] = «oa + &>V, [&>'] = to/3 + a)' ft',
respectively for the transformations Ak, B, C, D, we obtain the following results :
For the matrix (Ak) we have
[(i>r,i] = <0r,k, [<0r,k] = Vr,i, [<>>'r,i] = <0'r,k, [a>'r,k] = <»'r,i, 0" = 1, 2> • • • » P) >
or, in words, if 2a)r>i, 2&>'r>; be called the i-th pair of periods for the argument
ur, the change effected by the substitution A^ is an interchange of the first
and A;-th pairs of periods — no other change whatever being made.
When we are dealing with p quantities, the interchange of the first and £-th of these
quantities can be effected by a composition of the two processes (i) an interchange of the
first arid second, (ii) a cyclical change whereby the second becomes the first, the third
becomes the second, ..., thep-th becomes the Q»-l)-th, and the first becomes the p-tln.
Such a cyclical change is easily seen to be effected by the matrix
'0 1
^
1 0
0
ro
0 1
1 0
1 0.
0
X.
0
* Other sets of elementary matrices, by the multiplication of which any Abelian matrix can
be formed, can easily be chosen. One other obvious set consists of the matrices obtained by
interchanging the rows and columns of the matrices Ak, B, C, D.
t We may state the meaning of the matrices A*, B, C, D somewhat differently in accordance
with the property remarked in Ex. iii. , § 324.
333] BY COMPOSITION OF THESE FOUR. 555
which verifies the equations (IX.) § 324, for r=l. Hence the matrices A3, ..., Ap can
each be represented by a product of positive powers of the matrices E and Az. Thereby
the (jo + 2) elementary matrices A2, ..., Ap, B, C, D can be replaced by only 5 matrices E,
Az, B, C, D*.
Considering next the matrix B we obtain
[1 ' r ' n r n r / n / f* ^ •*•> ^> •••> ?^\
ft)nlj = ft)r)1, |G>r,iJ= — Wr,i, |wr,tj = <»r,ii Lft)r(rJ = <wr,i, I.. o 1,
\1 — Z, . . ., JP/
so that this transformation has the effect of interchanging m.r>l and &>',., i>
changing the sign of one of them ; no other change is introduced.
The matrix G gives the equation
[w'r.i] = 0>'r,i - «r,i, (r = 1, 2, . . . , p),
but makes no other change.
The matrix D makes only the changes expressed by the equations
[«•>, 1] = w 'r, i — d>r, 2 , [w'r, 2] = <u'r, 2 ~ &>r, i •
In applying these transformations to the case of the theta functions we
notice immediately that A^, C and D all belong to the case considered in
§ 326 (ii), in which the matrix a = 0.
Thus in the case of the transformation A^ we have
where w differs from u only in the interchange of u^ and uk, T differs from r
only in the interchange of the suffixes 1 and k in the constituents Tr>s of the
matrix T, and K, K' differ from Q, Q' only in the interchange of the first and
&-th elements both in Q and Q'. Thus in this case the constant A is equal
to 1.
In the case of the matrix (C), the equations of § 326 (2) give
where
u-w, T'=T save that r\tl=Tltl-I, and K'=Q', K=Q save that Ki = Ql
now the general term of the left-hand side, or
is equal to
r' (n+K')*+iir
_ g-tV (Q,'2- Q,') e2wiw (n+K')+iirr'(n+K"Jf>+2niK(n+K') .
thus in the case of the transformation (C) the constant A is equal to
g-i><Q,"-Q,') ; wnen Q/ is a half-integer, this is an eighth root of unity.
* See Krazer, Ann. d. Mat., Ser. n., t. xii. (1884). The number of elementary matrices is
stated by Burkhardt to be further reducible to 3, or, in case p = 2, to 2; Getting. Nachrichten
1890, p. 381.
556 DETERMINATION OF THE CONSTANT FACTOR [333
In the case of the matrix (D), the equations of § 326 (ii) lead to
where u = w, r —r save that r'1)2 = rli2 — 1, r'2)i = r2)1 — 1, and K' = Q', K = Q
save that Kl = Q1 + Q2', Kz = Q2 + Q/ ; now we have
riK (n+K1)
thus, in the case of the matrix (D) the constant A is equal to e~ZiriQi'Q*.
We consider now the transformation (B) — which falls under that con
sidered in (i) § 326. In this case Trio.' (a + TO?) wz is equal to irirltl w-?, and
the equation (a + TO?) r = J3 + r/3' leads to the equations
or, the equivalent equations (?^, s = 2, 3, . . . , p),
Iii /it _ / ''/'.
i.x /T" 1,1> ^"l,r — •,ttrl*1t\i ^"r,8 — *f,t T i,r T i,«/'r i,i>
also WI = TI)IWI, w;. = Tljr Wj + wr, so that Wj = — r'^iMj, wr = wr — r\>Tult and
Tl)1 Wj2 = - T'J,! Mj2 ; further we find
K' = Q' save that K,' =-Ql,andK=Q save that K-, = Q/ ;
with these values we have the equation
334. To determine the constant A in the final equation of the last
Article we proceed as follows* : — We have
ri
(i) e*wimwdw = 0 or 1,
Jo
according as m is an integer other than zero, or is zero ;
(ii) if a be a positive real quantity other than zero, and /9, 7, 8 be real
quantities,
where for the square root is to be taken that value of which the real part is
positive*f ;
* For indications of another method consult Clebsch u. Gordan, Abel. Funct., § 90; Thomae,
Crelle, LXXV. (1873), p. 224.
+ By the symbol *J n, where /* is any constant quantity, is to be understood that square root
whose real part is positive, or, if the real part be zero, that square root whose imaginary
part is positive.
335] FOR THESE FOUR TRANSFORMATIONS. 557
(iii) with the relations connecting u, w and r, r' given in the previous
Article,
un = (wn\ + (T!, l nt + + rlt p np) wl ,
where (wn\ denotes W2n2 + + wpnp ;
(iv) the series representing the function © (w, T') is uniformly con
vergent for all finite values of wlt ..., wp, and therefore, between finite limits,
the integral of the function is the sum of the integrals of its terms.
Therefore, taking the case when (?') and therefore (j£') are (°), and
integrating the equation
in regard to Wi, ... , wp, each from 0 to 1, we have
oo — oo, oo rl rl
Ao=nl=-«,n^,nl,L'"he7> n*dw1...dwp,
where, on the right hand, the integral is zero except for n2 = 0, . . . , np = 0 ;
thus
oo rl
n,= -oo Jo
v f1 i2
M1=-oo JO
J —ao
hence since the real part of irirltl is negative (§ 174), we have
A l^ I *
0 = A/ : = A/ — '
v 7rirltl v rlfl
where the square root is to be taken of which the real part is positive.
Hence
e»ir.,.wi«®(M; T) = A@(w; T')f
V T1>a
and from this equation, by increasing w by K+r'K , we deduce that
er*TIllWl«0(M; T Q')= /J.^.riQ.Q,'©^; T' ^
Hence, when the decomposition of any linear transformation into trans
formations of the form Ak, B, C, D is known, the value of the constant
factor, A, can be determined.
335. But, save for an eighth root of unity, we can immediately specify the value in
the general case ; for when Q, $ are zero, the value of the constant A has been found to
be unity for each of the transformations Ak, C, D, and for the transformation B to have a
558 THE CONSTANT FACTOR FOR ANY LINEAR TRANSFORMATION. [335
value which is in fact equal to *Ji/\M\, \M\ denoting the determinant of the matrix M.
Hence for a transformation which can be put into the form
, ......... ...... ....
a' £7 * *
if the values of the matrix M for these component transformations be respectively
...M^...\...\...M^.. .!...!...,
the value of the constant A, when Q, ty are zero, for the complete transformation, will be
but if the complete transformation give u = Mw, we have M=...M2M1...; thus, for any
transformation we have the formula
\M\
where M=a + ra, u = Mw, and e is an eighth root of unity, r, r being as in § 328, p. 544.
Putting 2o>M, 2vw for u, w, as in § 326, this equation is the same as
3 (u ; 2», 2»', 2V, 2,')= 7=== * U 5 2u, 2«', 2f, 2f ' | fl
/J
V|*
where | o> | is the determinant of the matrix w, etc.
Of such composite transformations there is one which is of some importance, that,
namely, for which
so that
Then
We may suppose this transformation obtained from the formula given above for the
simple transformation B — thus— Apply first the transformation B which interchanges
•"mi «'r,i with a certain change of sign of one of them; then apply the transformation
A 2BA2 which effects a similar change for the pair <or)2, a>'r>2 • then the transformation
A3£A3, and so on. Thence we eventually obtain the formula
Q
where
t T -i a .. . T o o
—J — I»ji 'f / &t Q
T 2, 2 — T2, 2 ) r .3, 3 — T 3, 3 ~i > • • • >
Tl, 1 T 2, 2
and, save for an eighth root of unity,
/~*~ / '~~i~ / i~ 1
where |T| is the determinant of the matrix T.
336] ELEMENTARY TRANSFORMATIONS OF PERIOD LOOPS. 559
The result can also be obtained immediately, and the constant obtained by an integra
tion as in the simple case of the transformation B ; we thus find, for the value of the
constant here denoted by x/ x/ ~ ..., the integral*
v Ti, i v T 2, 2
r ... r e™x*dxi---dxP-
J -CO J _CO
^!r. i. Prove that another way of expressing the value of this integral is
4tStan-i\r .
e r=i /V|TTO|,
where, if the matrix T be written p + icr, TTO| is the determinant of the matrix p2 + o-2,
which is equal to the square of the modulus of the determinant of the matrix T, also
A!, ..., \p are the (real) roots of the determinantal equation \p-\a-\ = 0, and tan"1 Ar lies
between - 7r/2 and w/2. Of the fourth root the positive real value is to be taken.
Ex. ii. For the case p = 1, the constant for any linear transformation is given by
according as a or a' is odd ; where a is positive, and
' i / 7r^a' •> I «•
as — a S — aa , T —. — «2 / i
L = e*<>- \/ / •
336. Returning now to consider the theory more particularly in con
nexion with the Riemann surface, we prove first that every linear trans
formation of periods such as
[to] = toot + 6>V, [a/] = w/3 + to'/S',
where
a£-£o = 0, «'£' - £'a' = 0, a0'-/3a' = l,
can be effected by a change in the manner in which the period loops are
taken. For this it is sufficient to prove that each of the four elementary
types of transformation, Ak, B, G, D, from which, as we have seen, every
such transformation can be constructed, can itself be effected by a change in
the period loops.
The change of periods due to substitutions Ak can clearly be effected
without drawing the period loops differently, by merely numbering them
* Weber has given a determination of the constant A for a general linear transformation by
means of such an integral, and thence, by means of multiple-Gaussian series. See Crelle, LXXIV.
(1872), pp. 57 and 09.
560
ELEMENTARY TRANSFORMATIONS.
[336
differently — attaching the numbers 1, k to the period-loop-pairs which were
formerly numbered k and 1. No further remark is therefore necessary in
regard to this case.
The substitution B, which makes only the change given by
[Q)ri J = ft)',., j , [ft/ r, i] = - ft),., ! ,
can be effected, as in § 320, by regarding the loop (6j) as an \a^\ loop, with
retention of its positive direction ; thus the direction of the (old) loop (a^),
which now becomes the [6J loop, will be altered ; the change is shewn by
comparing the figure of § 18 (p. 21) with the annexed figure (13).
FIG. 13.
The change, due to the substitution G, which is given by
[** r, i] = to'r, 1 ~ Mr, i >
is to be effected by drawing the loop [aj in such a way that a circuit of it
(which gives rise to the value [2ft/r,i] for the integral ur) is equivalent to a
circuit of the original loop («j) taken with a circuit of the loop (6j) from the
positive to the negative side of the original loop (a^.
This may be effected by taking the loop [aj as in the annexed figure (14)
(cf. § 331).
FIG. 14.
For the transformation D the only change introduced is that given by
[ft)'r, i] = ft>'r, i — &>r, 2 > [o>'r, 2] = «'»•, 2 ~ <»>•, 1 ,
and this is effected by drawing the loops \a^\, [a2], so that a circuit of
337]
OF PERIOD LOOPS.
561
[«j] is equivalent to a circuit of the (original) loop (a^ together with a
circuit of (62), in a certain direction, and similarly for [a2]. This may be
done as in the annexed diagram (Fig. 15).
FIG. 15.
For instance the new loop [«2] in this diagram (Fig. 15) is a deformation of a loop
which may be drawn as here (Fig. 16) ;
FIG. 16.
since the integrand of the Abelian integral ur is single-valued on the Riemann surface,
independently of the loops, the doubled portion from L to M is self-destructive ; and
a circuit of this new loop [a2] gives w'r> 2 - a>r> j , as desired.
Hence the general transformation can be effected by a composition of the
changes here given. It is immediately seen, for any of the linear transform
ations of § 326, that if the arguments there denoted by Ult ..., Up be a set
of normal integrals of the first kind for the original system of period loops,
then Wlt ..., Wp are a normal set for the new loops associated with the
transformation.
337. Coming next to the question of how the theory of the vanishing of
the Riemann theta function, which has been given in Chap. X., is modified
B. 36
562 TRANSFORMATION OF THE PLACES TO1} ...,mp. [337
by the adoption of a different series of period loops, we prove first that when
a change is made equivalent to the linear transformation
[eo] = <ya + a/a', [a/] = &>/3 + &//3',
the places m^, ..., mp of § 179, Chap. X., derived from any place TO, upon
which the theory of the vanishing of the theta function depends, become
changed into places TO/, . . . , mp' which satisfy the p equations
)], + J T;, , [d («'£')]! +... + lriip[d
wherein tt1} ..., t*p denote the normal integrals of the first kind for the
original system of period loops.
For let w1} ...,wp be the normal integrals of the first kind for the new
period loops, and let TO/, . . . , mp be the places derived from the place TO, in
connexion with the new system of period loops, just as m1} ..., mp were
derived from the original system. In the equation of transformation
e*ia' ,a+ra) vfi <H) ; T = A,® (w J T'),
put
= Wx'm —
so that the right-hand side of the equation vanishes when x is at any one of
the places TO/, ..., mp; then we also have
u =
hence the function
f , ; 3 (
l^(a/5)J
vanishes when x is at any one of the places acl) ..., xp\ therefore, by a
proposition previously given (Chap. X., § 184 (X.)), the places TO/, ..., TO/
satisfy the equivalence stated above.
It is easy to see that this equivalence may be stated in the form
It may be noticed also that, of the elementary transformations associated
with the matrices Ak, B, C, D, of § 333, only the transformation associated
with the matrix C gives rise to a change in the places mlf ..., mp\ for each
of the others the characteristic [^d(a/3), ^d(ct'/3')] vanishes.
338. From the investigation of § 329 it follows, by interchanging the
rows and columns of the matrix of transformation, that a linear trans-
339] CHARACTERISTICS OF RADICAL FUNCTIONS. 563
formation can be taken for which the characteristic [^d (ot/3), |d
represents any specified even characteristic; thus all the 2p-1(2p + 1) sets*,
/, ..., mp', which arise by taking the characteristic Jrj in the equivalence
to be in turn all the even characteristics, can arise for the places w/, . . . , mp'.
In particular, if ^fi^ M< be an even half-period for which 0 (i^M, /*) vanishes,
we may obtain for w/, ..., mp' a set consisting of the place m and p-l
places ??/, ..., n'p_i, in which w/, ..., yi'^-j are one set of a co-residual lot of
sets of places in each of which a ^-polynomial vanishes to the second order
(cf. Chap. X., § 185).
Ex. If in the hyperelliptic case, with jt> = 3, the period loops be altered from those
adopted in Chap. XI., in a manner equivalent to the linear transformation given in the
Example of § 329, the function e (w ; T'), denned by means of the new loops, will vanish
for w = 0; and the places mx', m2', m3', arising from the place a (§ 203, Chap. XL), as
mlf ...,mp arise from m in § 179, Chap. X., will consist t of the place a itself and two
arbitrary conjugate places, z and ~z.
339. We have, on page 379 of the present volume, explained a method
of attaching characteristics to root forms V.X(1), VF(3) ; we enquire now how
these characteristics are modified when the period loops are changed. It will
be sufficient to consider the case of VF(3) ; the case of */Xw arises (§ 244) by
taking <f>0*/Xw in place of VF(3). Altering the notation of § 244, slightly, to
make it uniform with that of this chapter, the results there obtained are as
follows ; the form X® is a polynomial of the third degree in the fundamental
^-polynomials, which vanishes to the second order in each of the places
Alt ..., Azp-3, ml, ..., mp, where A1} ..., A2p_3 are, with the place m, the
zeros of a ^-polynomial <£0; the form F(3) is a polynomial, also of the third
degree in the fundamental ^-polynomials, which vanishes to the second order
in each of the places A1} ..., Azp_3, fj,1, ..., /^; if
g/T<ll+...+2P/TiiJ,), (i = 1,2,. ..,|>),
where ul} ..., up are the Riemann normal integrals of the first kind, the
characteristic associated with the form F(3) is that denoted by £K J ; andj
it may be defined by the fact that the function \/F(3)/VZ(3), which is single-
valued on the dissected Riemann surface, takes the factors (— I)?.', (— 1)?«
respectively at the t'-th period loops of the first and second kind.
Take now another set of period loops ; let m/, . . . , mp be the places
* Or lot of sets, when the equivalence has not an unique solution.
t Cf. the concluding remark of § 185.
t Integer characteristics .being omitted.
36—2
564 TRANSFORMATION OF THE CHARACTERISTICS [339
which, for these loops, arise as ??ij, ..., mp arise for the original set of period
loops; let Z{s) be the form which, for the new loops, has the same character
as has the form X(*} for the original loops, so that Z® vanishes to the second
order in each of Al} ..., A2p-3> m/, ..., mp ; then from the equivalences
(§ 337)
*' mp s * [d (0/3% + K<,i [d (fiaOl + • • • + i r it p (d (aa')]p,
where wlt ..., wp are the normal integrals of the first kind, it follows, as in
§ 244, that the function V/£(3) /VJT(3) is single-valued on the Riemann surface
dissected by the new system of period loops, arid at the r-th new loops,
respectively of the first and second kind, has the factors
e-iri[d (oa'( ~\r Qiri [d (J3|3') ],._
The equations of transformation,
[<w] = &)« + &>V, [&>'] = w/3 + o)'/3',
of which one particular equation is that given by
express the fact (cf. § 322) that a negative circuit of the new loop [br] is
equivalent to ai>r negative circuits of the original loop (bi) and di<r positive
circuits of the original loop (o^) ; thus a function which has the factors e~™qt,
e^i at the i-th original loops, will at the r-th new loop [ar] have the factor
e—*Ur \ where lr' is an integer which is given by
- lr' = S [- qi «i, r + qt <*'i, r], (mod. 2) ;
i=l
thus the factors of VF(3)/VX(3) at the new period loops are given by e~vil/,
e1"1, where I, I' are rows of integers such that
I' = aq' - a'q, -l = J3qf- ft'q, (mod. 2).
Therefore the factors of \iT®/^Z1® = (VF^Vl/^^/VZ^Xat the
new period loops, are given by e~irik>, e™*, where
k' = aq' -aq-d (aax), -k = P<f-ffq-d (yS/97), (mod. 2) ;
now the characteristic associated with VF(3) corresponding to the original
system of period loops may be defined by the factors of VF(3)/VjT<3> at those
loops; similarly the characteristic which belongs to VF(3) for the new system
_ _ /£'\
of loops is defined by the factors of \/F(3) /*JZ®, and is therefore if.,); the
equations just obtained prove therefore that the characteristic associated with
VF(3) is transformed precisely as a theta characteristic.
340] OF RADICAL FUNCTIONS. 565
The same result may be obtained thus ; the p equations of the form
are immediately seen, by means of the equation (a +TO.') (&' — r'd') = 1 to lead to p equations
expressible by
if" m' 4- . . .
subtracting from these the equations
(i=},...,
we obtain equations from which (as in § 244) the characteristic of V 'F<3>, for the new
loops, is immediately deducible.
Similar reasoning applies obviously to the characteristics of the forms
considered on page 380 (§ 245). But the characteristic for a form
(p. 381), which is obtained by consideration of the single-valued
function ^JT***/^W — into which the form *JX®, depending on the places
m1) ..., mp, does not enter — is transformed in accordance with the equations
k' = aq -aq, -k = J3q' - J3'q, (mod. 2),
and may be described as & period- characteristic, as in § 328.
340. Having thus investigated the dependence of the characteristics
assigned to radical forms upon the method of dissection of the Riemann
surface, it is proper to explain, somewhat further, how these characteristics
may be actually specified for a given radical form. The case of a form
\/Jf w differs essentially from that of a form VJT(2"+1>. When the zeros of a
form \fX(<i>i) are known, and the Riemann surface is given with a specified
system of period loops, the factors of a function VX(a't)/3>('t) at these loops
may be determined by following the value of the function over the surface,
noticing the places at which the values of the function branch — which places
are in general only the fixed branch places of the Riemann surface ; the
process is analogous to that whereby, in the case of elliptic functions, the
values of Vp (u + 2^) - ^ / \/p (u) - el , Vp (u + 2&>o) - ej^p (u) - ^ may be
determined, by following the values of Vp (u) - el over the parallelogram of
periods. But it is a different problem to ascertain the factors of the function
\/F(3)/vAA>) at the period loops, because the form VZ(3) depends upon the
places MI, ..., mp, and we have given no elementary method of determining
these places ; the geometrical interpretation of these places which is given in
§ 183 (Chap. X.), and the algebraic process resulting therefrom, does not
distinguish them from other sets of places satisfying the same conditions;
the distinction in fact, as follows from § 338, cannot be made algebraically
unless the period loops are given by algebraical equations. Nevertheless we
566 DETERMINATION OF THE PLACES m1} ...,mp [34-0
may determine the characteristic of a form F(3), and the places m^ ..., mp,
by the following considerations*: — It is easily proved, by an argument like
that of § 245 (Chap. XIII.), that if there be a form VX(1) having the same
characteristic as \/F(3), there exists an equation of the form *JX(l] \/F(3) = <E>(2) ;
and conversely, if q + 1 linearly independent polynomials, of the second
degree in the p fundamental ^-polynomials, vanish in the zeros of VF(3), and
M^2* denote the sum of these q + 1 polynomials, each multiplied by an
arbitrary constant, that we have an equation \/F(1) \^F®»1!*', where VF(1) is
a linear aggregate of q + 1 radical forms like *JX(l), all having the same
characteristic as VlT(s) ; in general, since a form M^2' can contain at most
3 (p — 1) linearly independent terms (§ 111, Chap. VI.), and the number of
zeros of VF(3) is 3 (p — 1), we have ^ + 1=0; in any case the value of q + 1
is capable of an algebraic determination, being the number of forms 4>('2)
which vanish in assigned places. Now the number of linearly independent
forms VjT(l) with the same characteristic is even or odd according as the
characteristic is even or odd (§§ 185, 186, Chap. X.) ; hence, without deter
mining the characteristic of VF(3) we can beforehand ascertain whether it is
even or odd by finding whether q + 1 is even or odd. Suppose now that
fa, ..., fj,p and fa, ..., fjLp' are two sets of places such that
m being an arbitrary place, and m, A1} ..., A2p_3 being the zeros of any
^-polynomial </>0 ; so that /^ , . . . , pp and /V, •••> /V are two sets arbitrarily
selected froni 2'2p sets which can be determined geometrically as in § 183,
Chap. X. (cf. § 244, Chap. XIII.) ; let F(3) vanish to the second order in each
of yLtj, ..., /Ap, A1} ..., A2p_3 and F/3' vanish to the second order in each of
pi> •••, Pp, AI, ..., A2p_3; by following the values of the single-valued
function VF^ys/F^ on the Riemann surface, we can determine its factors at
the period loops ; at the r-th period loops of the first and second kind let
these factors be (—I)*'-', (— !)*'• respectively; then if ^(qit ..., <&,') and
2" (Qi> •••• Qp) be respectively the characteristics of VF(3) and VF/3', which we
wish to determine, we have (§ 244)
krf = Qr' - qr', kr = Qr - qr, (mod. 2).
Take now, in turn, for /*/, ..., /A/, all the possible 2^ sets which, as in § 183,
are geometrically determinate from the place m; and, for the same form
VF(3), determine the 2'* characteristics of all the functions VF^'/VF*3' arising
* Noether, Jahresbericht der Deutschen Mathematiber Vereinigung, Bd. iii. (1894), p. 494,
where the reference is to Fuchs, Crelle, LXXIII. (1871) ; cf. Prym, Zur Theorie der Fumtionen in
einer zweibldttrigen Fldche (Zurich, 1866).
341] WHEN THE PERIOD LOOPS ARE GIVEN. 567
by the change of the forms \f Y^ ; then there exists one, and only one,
s'\
} , satisfying the condition that the characteristic
_
is even when VF/3' has an even characteristic and odd when VFj(3) has an odd
characteristic ; for, clearly, the characteristic \ ( 1 is a value for \ ( ] which
\q / \s /
satisfies the condition, and if i I I were another possible value for 1- ( 1
\<r ) \s)
we should have
(k + o-) (k' + </) = (k + q)(k + q') (mod. 2),
or
k (<r' - q) + k' (<r-q) = qq' - <ra'
for all the 22^ possible values of |- , ; and this is impossible (Chap. XVII.,
\fc 1
§ 295).
Hence we have the following rule : — Investigate the factors of V F^/V F(3)
for an arbitrary form V F(3) and all 2^ forms V F^3' ; corresponding to each
form VFj'3' determine, by the method explained in the earlier part of this
Article, whether its characteristic is even or odd ; then, denoting the factors of
any function \/Fj(3)/v F(3) respectively at the first and second kinds of period
loops by quantities of the form (— !)*,(— 1)*, determine the characteristic M ),
satisfying the condition that the characteristic A- ( , 1 is, for every form
\ + k ) y J
j'3', even or odd according as the characteristic of that form, VF/3', is even or
odd; then ^ ( } is the characteristic of the form VF(3); this being determined
the characteristic of every form VF^ is known; the particular form ^Y^ for
which the characteristic, thus arising, is actually zero, is the form previously
denoted by VZ(3> — namely the form vanishing in the places ml} ..., mp which are
to be associated (as in § 179, Chap. X.) with the particular system of period
loops of the Riemann surface which has been adopted.
Thus the method determines the places m1} ...,mp and determines the
characteristic of every form \/F(3); the characteristic of any other form
v/F(2l/+1) is then algebraically determinable by the theorems of § 245 (p. 380).
341. For the hyperelliptic case we have shewn, in Chap. XL, how to
express the ratios of the 2^ Riemann theta functions with half-integer
characteristics by means of algebraic functions ; the necessary modification
568 APPLICATION OF THE THEORY [341
of these formulae when the period loops are taken otherwise than in
Chap. XL, follows immediately from the results of this chapter. If the
change in the period loops be that leading to the linear transformation
which is associated with the Abelian matrix formed with the integer
matrices a, /3, a', ft', we have (§ 324)
where
k' = aq' -a'q-d (aa'), - k = J3q' - 0'q -
If now, considering as sufficient example the formula of § 208 (Chap. XL), we
have
ur a = ?l6)M + •" + <fpt°r,p + qi'w'r.i + »• + Qp' O>r,p,
then we have
Wr'a= llVr>1+ ... + lpVr,p + liVrtl + ... +lpVfip,
where
l' = aq'-a'q = k'+d(aa'), - I = 0q' - /3'q = - k + d(W) ;
therefore, if the characteristic \ (d (/9/3'), d (aa')) be denoted by p, the function
&r 1 40 ; £ ( , ) I is a constant multiple of ^x \w ; \ ( , ) + p ; and we may
\/c / \ \*/
denote the latter function by ^ [w\wb'a + fi\. Thus the formula of § 208 is
equivalent to
- \ — 71 - ~\ n ^i (w I w*' a + /*)
,...6-=
where C is independent of the arguments w1} ..., wp, and, as in § 206,
Wr = <""' + . .. + <"<**, (r = 1,2,.. .,/)).
Similar remarks apply to the formula of §§ 209, 210. It follows from
§ 337 that the characteristic fj, is that associated with the half-periods
where m^, ..., mp' are the places which, for the new system of period loops,
play the part of the places ml} ..., mp of § 179, Chap. X. It has already
(§ 337) been noticed that for the elementary linear substitutions Ak, B, D the
characteristic yu, is zero.
342. In case the roots clt alt ca, «2> •••> c, in the equation associated with
the hyperelliptic case
7/2 _ 4 (x _ Cj) ^ _ tti) ^ _ Cz) (x — at)...(x- Cp) (x - Op) (x - c),
be real and in ascending order of magnitude, we may usefully modify the
notation of § 200, Chap. XI. Denote these roots, in order, by byp, 62p_i, . .., b0,
342]
TO THE HYPERELLIPTIC THETA FUNCTIONS.
569
so that bti, byi-i are respectively cp_t+i, ap-i+i and b0 is c, and interchange
the period loops (a;), (£>;), with retention of the direction of (&{), as in the
figure annexed (Fig. 17).
FIG. 17.
Then if U*' a, ..., Ux< a are linearly independent integrals of the first kind,
such that dUxr'a ldx = -^r/y, where tyr is an integral polynomial in x, of degree
p — 1 at most, with only real coefficients, the half-periods
are respectively real and purely imaginary, so that [a/,.,;] is also purely
imaginary ; if now w*' a, ..., w^' a be the normal integrals, so that
then the second set of periods of w*'a, ..., wx'a, which are given by
T'r,i = Lr<l[2a>\!i] + ...+Lr>p[2a>'p>i], (r,s=l, 2, ...,p\
are also purely imaginary* ; forming with these the theta function ® (w ; T'),
the theta function of Chap. XI. is given (§ 335) by
e-in* @ (u. T Q') = AeMQQ> ®(w.r>\ -x)t
where K, K' are obtainable from Q, Q' respectively by reversing the order
of the p elements, and A is the constant Vt/A7 Vt'Si/A^ Vt'A2/ A3 . . . , in which
^ = TI,I, A2 = T^T^ — T21)2, etc. We find immediately that
(i = (), 1, .... p), and may hence associate with &„•_,, 62i the respective odd and
even characteristics
vo; v- 1
* The quantities rt., of Chap. XI. (of which the matrix is given in terms of the r'< of § 342
by rr' = - 1) are also purely imaginary when c, , a,, ..., cp, ap, c are real and in ascending order
of magnitude.
570 WEIERSTRASS'S RULE. [342
and may denote the theta functions with these characteristics respectively by
@2t-i(w; T'), ©«(w; T'); if bk, blt bm, ..., be any of the places b.2p, ..., 60, not
more than p in number, and if, with 0 }> qt < 2, 0 > g/ < 2, we have
then the function whose characteristic is A- ( * J may be denoted by
-
This function is equal to, or equal to the negative of, the function with
characteristic J ( ) j according as the characteristic is even or odd.
We have thus a number notation for the 22p half-integer characteristics*,
equally whether the surface be hyperelliptic or not ; this notation is under
stood to be that of Weierstrass (Kb'nigsberger, Crelle, LXIV. (1865), p. 20).
For the numerical definition of the half-periods, which are given by the rule
at the bottom of p. 297, precise conventions are necessary as to the allocation
of the signs of the single valued functions V# — br on the Riemann surface
(cf. Chap. XXII.).
In the hyperelliptic case j» = 2, the characteristics of the theta functions given in the
table of § 204 are supposed to consist of positive elements less than unity ; when Q1 , Q2 ,
Qi, Q2' are each either 0 or ^, the formula of the present article gives
«"*"• 9 \u ; r \ y Vl^Ae-MW 6 \w ; r'
**1 % J I-
the number notations for the transformed characteristics are then immediately given by
the table of § 204. The result is that the numbers
02, 24, 04, 1, 13, 3, 5, 23, 12, 2, 01, 0, 14, 4, 34, 03
are respectively replaced by
3, 1, 13, 24, 04, 02, 5, 0, 4, 2, 34, 23, 14, 12, 01, 03.
* For convenience in the comparison of results in the analytical theory of theta functions, it
appears better to regard it as a notation for the characteristics rather than for the functions.
344]
CHAPTER XIX.
ON SYSTEMS OF PERIODS AND ON GENERAL JACOBIAN FUNCTIONS.
343. THE present chapter contains a brief account of some general ideas
which it is desirable to have in mind in dealing with theta functions in
general and more especially in dealing with the theory of transformation.
Starting with the theta functions it is possible to build up functions
of p variables which have 2p sets of simultaneous periods — as for instance
by forming quotients of integral polynomials of theta functions (Chap. XL,
§ 207), or by taking the second differential coefficients of the logarithm of
a single theta function (Chap. XL, § 216, Chap. XVIL, § 311 (8)). Thereby
is suggested, as a matter for enquiry, along with other questions belonging to
the general theory of functions of several independent variables, the question
whether every such multiply-periodic function can be expressed by means of
theta functions*. Leaving aside this general theory, we consider in this
chapter, in the barest outline, (i) the theory of the periods of an analytical
multiply-periodic function, (ii) the expression of the most general single
valued analytical integral function of which the second logarithmic dif
ferential coefficients are periodic functions.
344. If an uniform analytical function of p independent complex
variables ul} ...,up be such that, for every set of values of u^, ...,up, it
is unaltered by the addition, respectively to u1}...,up, of the constants
PH ..., Pp, then Pj, ..., Pp are said to constitute a period column for the
function. Such a column will be denoted by a single letter, P, and Pk will
denote any one of Plt ..., Pp. It is clear that if each of P, Q, R, ... be
period columns for the function, and X, p, v, ... be any definite integers,
independent of k, then the column of quantities \Pk + (jbQk + vRk + ... is
also a period column for the function ; we shall denote this column by
\P + pQ + vR+ ..., and say that it is a linear function of the columns
P, Q, R, ..., the coefficients X, p, v, ..., in this case, but not necessarily
* Cf. Weierstiass, (Jrelle, LXXXIX. (1880), p. 8.
572 CONDITIONS THAT AN UNIFORM FUNCTION [344
always, being integers. The real parts of the new column are the same
linear functions of the real parts of the component columns, as also are the
imaginary parts. More generally, when the p quantities XP* + pQk + vRk + • • •
are zero for the same values of X, /JL, v, . . . , we say that the columns P,Q,R,...
are connected by a linear equation ; it must be noticed, for the sake of
definiteuess, that it does not thence follow that, for instance, P is a linear
function of the other columns, unless it is known that X is not zero.
It is clear moreover that any 2p + l, or more, columns of periods are
connected by at least one linear equation with real coefficients (that is, an
equation for each of the p positions in the column — p equations in all, with
the same coefficients) ; for, in order to such an equation, the separation of
real and imaginary gives 2p linear equations to be satisfied by the 2p 4- 1
real coefficients ; allowing possible zero values for coefficients these equations
can always be satisfied.
For instance the periods Q — Qj^ + iQ^, o) = <a1 + t<»2> w'^wj' + iwg', are connected by an
equation
in which however, if a>1a>2' — ^ta^O, also £ = 0.
Thus, for any periodic function, there exists a least number, r, of period
columns, with r lying between 1 and 2p + 1, which are themselves not
connected by any linear equation with real coefficients, but are such that
every other period column is a linear function of these columns with real
finite coefficients. Denoting such a set* of r period columns by P(1), ...,P(r),
and denoting any other period column by Q, we have therefore the p
equations
XrP<r)
wherein X1? ..., Xr are independent of k, and are real and not infinite. It is
the purpose of wharf follows to shew, in the case of an uniform analytical
function of the independent complex variables ult ..., up, (I.) that unless the
function can be expressed in terms of less than p variables which are linear
functions of the arguments ul} ..., up, the coefficients \ly . . . , X,. are rational
numbers, (II.) that, Xls ...,X,. being rational numbers, sets of r columns of
periods exist in terms of which every existing period column can be linearly
expressed with integral coefficients.
Two lemmas are employed which may be enunciated thus : —
(a) If an uniform analytical function of the variables ult ..., up have a
column of infinitesimal periods, it is expressible as a function of less than
p variables which are linear functions of w1, ..., up. And conversely, if such
* It will appear that the number of such sets is infinite ; it is the number r which is unique.
t These propositions are given by Weierstrass. Abhandlungen am der Functionenlehre
(Berlin, 1886), p. 165 (or Berlin. Monatsber. 1876).
345] HAVE INFINITESIMAL PERIODS. 573
uniform analytical function of ul} ..., uv be expressible as a function of less
than p variables which are linear functions of ull ..., up, it has columns of
infinitesimal periods.
(#) Of periods of an uniform analytical function of the variables
•ult...,up, which does not possess any columns of infinitesimal periods,
there is only a finite number of columns of which every period is finite.
345. To prove the first part of lemma (a) it is sufficient to prove that
when the function f(ui, ..., up) is not expressible as a function of less than
p linear functions of ult ..., up, then it has not any columns of infinitesimal
periods.
We define as an ordinary set of values of the variables ult ...,up a set
Ui, ..., up', such that, for absolute values of the differences ut — «/, ...,up — Up'
which are within sufficient (not vanishing) nearness to zero, the function,
/(MI, ..., Up), can be represented by a converging series of positive integral
powers of these differences — the possibility of such representation being the
distinguishing mark of an analytical function ; other sets of values of the
variables are distinguished as singular sets of values*.
Then if the function be not expressible by less than p linear functions of
«!, ..., Up, there can exist no set of constants d, ..., cp such that the
function
vanishes for all ordinary sets of values of the variables; for this would
require / to be a function of the p-l variables c^ - c&i (i = 2, ..., p).
Hence there exist sets of ordinary values such that not all the differential
coefficients df/du,, ...,df(dup are zero; let <$\ ....u™ be such an ordinary
set of values; for all values of «,, ..., up in the immediate neighbourhoods
respectively of «, , ..., u™, the statement remains true that not all the partial
differential coefficients are zero.
Then, similarly, the determinants of two rows and columns formed from
the array
dup
do not all vanish for every ordinary set of values of the variables; let
*!,...,«*, be an ordinary set for which they do not vanish ; for all values of
* The ordinary sets of values constitute a continuum of 2p dimensions, which is necessarily
limited; the limiting sets of values are the singular sets. Of. Weierstrass, Crelle LXXXIX
1880 . 3.
(1880), p. 3.
574 INFINITESIMAL PERIODS. [345
ul} ...,up in the immediate neighbourhoods respectively of u? , ..., U* , the
statement remains true that not all these determinants are zero.
Proceeding step by step in the way thus indicated we infer that there exist
sets of ordinary values of the variables, (ul\ ..., u(p), . . . , (u\p\ . . . , •z/jf)), such
that the determinant, A, of p rows and columns in which the k-ih element of
the r-th row is df(u(i\ ..., u^^/du^, does not vanish; and since these are
ordinary sets of values of the arguments, this determinant will remain
different from zero if (for r=l, . .., p) the set u(i\ ..., u(p be replaced
by vf , ..., Vp, where t^ is a value in the immediate neighbourhood of
(r)
%•
This fact is however inconsistent with the existence of a column of
infinitesimal periods. For if Hl} ..., Hp be such a column, of which the
constituents are not all zero, we have
f TT J r,,w _L A TT Jr) . a 77 1
= k=i du L ' + l " '"' p + WW
where #1} ..., 0p are quantities whose absolute values are ^ 1, and the
bracket indicates that, after forming df/duk, we are (for ra=l, . .., p) to
substitute urm + 6mHm for u^ ; these p equations, by elimination of Hl> ... , 5^
give zero as the value of a determinant which is obtainable from A by slight
changes of the sets u[ , ...,Up ; we have seen above that such a determinant
is not zero.
To prove the converse part of lemma (a) we may proceed as follows.
Suppose that the function is expressible by m arguments vl} ..., vm given by
Vk = aktlu1+ ... + akt pup, (k = 1, . . ., m),
wherein m< p. The conditions that v1}...,vm remain unaltered when
MI, ..., up are replaced respectively by u1 + tQi, ..., up + tQp are satisfied by
taking Q1} ..., Qp so that
and since m<p these conditions can be satisfied by finite values of Qlt ...,QP
which are riot all zero. The additions of the quantities tQi,...,tQp to
HI, ...,up, not altering vlt ..., vm, will not alter the value of the function f.
Hence by supposing t taken infinitesimal ly small, the function has a column
of infinitesimal periods.
346. As to lemma (@), let Pk = pk + ia-k be one period of any column of
periods, (k=I, ..., p), wherein pk, <rk are real, so that, in accordance with the
condition that the function has no column of infinitesimal periods, there
347] A SYSTEM OF INDEPENDENT PERIODS. 575
is an assignable real positive quantity e such that not all the 2p quantities
pk, ak are less than e. Then if fit, vk be 2p specified positive integers,
there is at most one column of periods satisfying the conditions
<(/**+l)e, vke$\<rk\<(vk + l)c, (k = 1, ..., p) ;
wherein \pk\, \<fk\ are the numerical values of pk, crk; for if pk + i<rk were
one period of another column also satisfying these conditions, the quantities
pk' — pk -|- { (0-j/ _ o-j.) would form a period column wherein every one of the
2p quantities pk — pk> a-k — o~k was numerically less than e.
Hence, since, if g be any assigned real positive quantity, there is only a
finite number of sets of 2p positive integers /jik, vk such that each of the
2p quantities pke, rke is within the limits (—g, g), it follows that there
is only a finite number of columns of periods Pk = pk + i(rk, such that each of
pk, <rk is numerically less than g. And this is the meaning of the lemma.
347. We return now to the expression (§ 344) of the most general
period column of the function /by real linear functions of r period columns,
of finite periods, in the form
here the suffix is omitted, and we make the hypothesis that the function
is not expressible by fewer than p linear combinations of ult ..., up.
Consider, first, the period columns Q from which X2 = X3 = ... = Xr = 0
and 0 < A! :}> 1. Since there are no columns of infinitesimal periods, there
is a lower limit to the values of \ corresponding to existing period columns
Q satisfying these conditions ; and since there is only a finite number of
period columns of wholly finite periods, there is an existing period for which
Xj is equal to this lower limit. Let Xltl be this least value of \1} and Q(1)
be the corresponding period column Q.
Consider, next, the period columns Q for which X3 = \4 — . . . = Xr = 0,
and O^-Xj^-l, 0 < X-j ^ 1. As before there are period columns of this
character in which X2 has a least value, which we denote by X2) 2. If there
exist several corresponding values of X1} let X1)2 denote one of these, and
denote \1>2PW + X2)2P<2> by Q<2'.
In general consider the period columns of the form
wherein
Since there are no infinitesimal periods, there is a lower limit to the values
of \m corresponding to existing period columns satisfying these conditions ;
since there is only a finite number of period columns of wholly finite periods,
there is at least one existing column Q for which X™ is equal to this lower
576 EXPRESSION OF ANY PERIOD [347
limit; denote this value of X™ by Xm,w, and denote by X1(TO, ..., \m-i,m values
arising in an actual period column Q(w) given by
Q<™> = X1)WP<1> + X2,mP<2> + ... + \m,mP(m> 5
there may exist more than one period column in which the coefficient of
P<«> is Xw,m.
Thus, taking m = l, 2, ..., r, we obtain r period columns Q*1*, ..., Q(r).
In terms of these any period column Q, = XjP'1' + ... + XrP(r>, in which
Xj, ..., Xr are real, can be uniquely written in the form
wherein Nlt ..., Nr are integers, and ^, ..., /ir are real quantities which are
zero or positive and respectively less than \l>1, ..., \r,r- For, putting Xr into
the form Nr\r,r + Pr, where Nr is an integer and fir, if not zero, is positive
and less than \rtr, we have
where
Xj = Xj — lv,.X1)r, . . . , X r_j = Xr_j IV r \r—i>r ,
and herein the column Q' = X1/P<1) + ... + X/r_iP(r~1) can quite similarly be
expressed in the form
and so on.
But now, if AW + ... + ^rQ(r) +/*iP(1) + ... + P<rP(r) be a period column,
it follows, aaNlt ...,Nr are integers, that also ^Pw +...+ /^rP(r> is a period
column; and this in fact is only possible when each of ^, ..., /j,r is zero.
For, by our definition of Q(r), the coefficient fj,r is zero ; then, by the definition
of Q(r~l), the coefficient /*,._! is zero ; and so on.
On the whole we have the proposition (II., § 344) — if
Qw =X1,mP<1» + ... +Xm,mP<™>, (m = l, ..., r),
be that real linear combination of the first in columns from P(1>, ..., P(r] in
which the m-th coefficient \m, m has the least existing value greater than zero
and not greater than unity, or be one such combination for which \m,m satisfies
the same condition, then every period column is expressible as a linear combina
tion of the columns Q(1), ..., Q(r) with integral coefficients.
It should be noticed that #*), ..., $r> are not connected by any linear equation with
real coefficients, or the same would be true of PW, ..., &rl And it should be borne
in mind that the expression of any period column by means of integral coefficients,
in terms of QW, ..., tyr\ is a consequence of the fact that the function /(MI} ..., up)
has only a limited number of period columns which consist wholly of finite periods.
Conversely the period columns, of finite periods, obtainable with such integral coefficients,
are limited in number,
348] BY MEANS OF INDEPENDENT PERIODS. 577
Another result (I., § 344) is thence obvious — The coefficients in the linear
expression of any period column in terms of P(1), ..., P(r) are rational
numbers.
For by the demonstration of the last result it follows that the period
columns P(1), ..., P{r} can be expressed with integral coefficients in terms of
Q'1', ..., Q(r) in the form
P<™> = N("l]Q(l} + ... + N^Qv, (m= 1, ..;,*•) ;
from these equations, since the columns P(1), ..., P(r) are not connected by
any linear relation with real coefficients, the columns Q(1), ..., Q(r) can be
expressed as linear combinations of P(1), ..., P(r) with only rational numbers
as coefficients; hence any linear combinations of Qw, ..., Q(r) with integral
coefficients is a linear combination of P(1), ..., P(r} with rational-number
coefficients.
It needs scarcely* to be remarked that the set of period columns
Q(l), ..., Q(r}, in terms of which any other column can be expressed with
integral coefficients, is not the only set having this property.
348. We consider briefly the application of the foregoing theory to the case of uniform
analytical functions of a single variable which do not possess any infinitesimal periods. It
will be sufficient to take the case when the function has two periods which have not a real
ratio ; this is equivalent to excluding singly periodic functions.
If 2(0!, 2«2 be two periods of the function, whose ratio is not real, and 2Q be any other
period, it is possible to find two real quantities Xx, X2 such that
Q = X1o>1 -|-X2&}2 ;
then of periods of the form 2X1o>1, in which 0<X1^>1, of which form periods do exist, 2<a1
itself being one, there is one in which Xj has a least value, other than zero — as follows
because the function has only a finite number of finite periods. Denote this least value
by /ij, and put Ql=p.1tal. Of periods of the form 2X1o>1 + 2X2w2 in which G^>X! ^>1, 0<X2^>1,
there is a finite number, and therefore one, in which X2 has the least value arising, say /z2;
let one of the corresponding values of X, be X ; put Q2 = Xa>1 + /i2<B2- Then any period
2Q = 2X1&>1 + 2X2o>2 is of the form 2NlQl + 2N2Q2 + 2v1o>1 + 2v<2<o2, where vlt v2 are (zero or)
positive and respectively less than ^ and p2, and jYj, Nz are integers, such that X2 = -/V2/i2 + »'2>
Xj — N^ = Nlpl-\-vl. But the existence of a period Q-2iV1Q1-2iV2G2 = 2j/1a>1 + 2i/2<B2 with
!/!</*!, v%<pi is contrary to the definition of /^ and ^2, unless vv and i>.2 be both zero.
Hence every period is expressible in the form . «
where N^ N2 are integers.
In other words, a uniform analytical function of a single variable without infinitesimal
periods cannot be more than doubly periodic^.
* For the argument compare Weierstrass (1. c., § 344), Jacobi, Ges. Werke, t. ii., p. 27,
Hermite, Crelle, XL. (1850), p. 310, Biemarm, Crelle, LXXI. (1859) or Werke (1876), p. 276. See
also Kronecker, "Die Periodensysteme von Functionen reeller Variabeln," Sitzungsber. der
Berl. Akad., 1884, (Jun. bis Dec.), p. 1071.
t Cf. Forsyth, Theory of Functions (1893), §§ 108, 107. It follows from these Articles, in
this order, that any three periods of a uniform function of one variable can be expressed, with
B. 37
OF THK
UNIVERSITY
578 BEARINGS OF THE THEORY. [348
It follows also that every period is expressible by 2o>l, 2o>2 with only rational-number
coefficients.
349. Ex. i. If r quantities be connected by k homogeneous linear equations with
integral coefficients (r>k\ it is possible to find r - k other quantities, themselves expressible
as linear functions of the r quantities with integral coefficients, in terms of which the r
quantities can be linearly expressed with integral coefficients.
Ex. ii. If PW, ... , PM be r columns of real quantities, each containing r - 1 constituents,
a column JVlP(l) + ...+JVrP^r) can be formed, in which Nlt ..., Nr are integers, whose r-l
constituents are within assigned nearness of any r — 1 assigned real quantities (cf.
Chap. IX., § 166, and Clebsch u. Gordan, Abels. Funct., p. 135).
Ex. m. An uniform analytical function of p variables, having r period columns P*1),
..., PW, each of p constituents, and having a further period column expressible in the
form X1P<1) + ...+XrP<r), wherein X1} ..., Xr are real, will necessarily have a column of
infinitesimal periods if even one of the coefficients X1} ... , Xr be irrational.
From this result, taken with Ex. i., another demonstration of the proposition of the
text (§ 347) can be obtained. The result is itself a corollary from the reasoning of the
text.
Ex. iv. If 7/j'a, ..., ux'a be linearly independent integrals of the first kind, on a
Riemann surface, and the periods, 2a>r,g, 2w'r,8, of the integral u*' a be written pr,,-H'oy,,,
p'^ + iV,.,,, shew that the vanishing of the determinant of 2p rows and columns which is
denoted by
°V,i> •••> °V, PJ °>,i> •••> °"r,p
would involve* the equation
( Ml — iNj) u*' a + . . . + (Mp — iNp} ux> a = constant,
where Ml, Nlt ..., Mp, JVP are the minors of the elements of the first column of this
determinant and are supposed not all zero.
The vanishing of this determinant is the condition that the period columns of the
integrals should be connected by a homogeneous linear relation with real coefficients.
350. The argument of the text has important bearings on the theory of the Inversion
Problem discussed in Chap. IX. The functions by which the solution of that problem is
expressed have 2p columns of periods in terms of which all other period columns can be
expressed linearly with integral coefficients ; these %p columns are not connected by any
linear equation with integral coefficients (§ 165), and, therefore, are not connected by any
linear equation with real coefficients.
It has been remarked (§ 174, Chap. X.) that the Riemann theta functions whereby the
2j9-fold- periodic functions expressing the solution of the Inversion Problem can be built
up, are not the most general theta functions possible. The same is therefore presumably
true of the 2/?-fold periodic functions themselves. Weierstrass has stated a theorem t
integral coefficients, in terms of two periods. These two periods, and any fourth period of the
function, can, in their turn, be expressed integrally by two other periods — and so on. The
reasoning of the text shews that when the function has no infinitesimal periods, the successive
processes are finite in number, and every period can be expressed, with integral coefficients,
in terms of two periods.
* Forsyth, Theory of Functions (1893), p. 440, Cor. ii.
t Berlin, Monatsber. Dec. 2, 1869, Crelle, LXXXIX. (1880). For an application to integrals
of radical functions, Cf. Wirtinger, Untersuchungen ilber Thetafunctionen (Leipzig, 1895), p. 77.
351]
DEFINITION OF GENERAL JACOBIAN FUNCTION.
579
whereby it appears that the most general 2p-fold periodic functions that are possible can
be supposed to arise in the solution of a generalised Inversion Problem ; this Inversion
Problem differs from that of Jacobi in that the solution involves multiform periodic
functions*; the theorems of the text possess therefore an interest, so far as they
hold, in the case of such multiform functions. The reader is referred to Weierstrass,
Abhandlungen am der Functionenlehre (Berlin, 1886), p. 177, and to Casorati, Acta
Mathematica^ t. viii. (1886).
351. We pass now to a brief account of a different theory which is
necessary to make clear the position occupied by the theory of theta
functions. Considering, a priori, uniform integral analytical functions
which, like the theta functions, are such that their partial logarithmic
differential coefficients of the second order are periodic functions, we in
vestigate certain relations which must necessarily hold among the periods,
and we prove that all such functions can be expressed by means of theta
functions.
Suppose that to the p variables u^, ...,up there correspond a- columns of
quantities af(i=l, ..., p, j = 1, ..., a) and a columns of quantities &(/>—
according to the scheme
a
"
and suppose </>(V) to be an uniform, analytical function of Wj, ..., up which
for finite values of ult ..., up is finite and continuous — which further has the
property expressed by the equations
<£ (U + d®) = e*r»W[«+JaM]+2«*<>» 0 (U\ (j=l,...,a), (I.)
wherein fttf) is a symbol for a column 6<J), ..., by and c'^ is a single quantity
depending only on j. The aggregate of c(1),...,c«" will be called the
characteristic or the parameter of <f>(u); af will finally be denoted by aitj.
We suppose that the columns a^ are independent, in the sense that there
exists no linear equation connecting them of which the coefficients are
rational numbers; but it is not assumed that the columns a® constitute all
the independent columns for which the function $ satisfies an equation of
the form (I.). Also we suppose that the equation (I.) is not satisfied for
any column of wholly infinitesimal quantities put in place of a&. The
reason for this last supposition is that in such case it is possible to express
</> as the product of an exponential of a quadric function of ul} ..., up,
multiplied into a function of less than p variables, these fewer variables
being linear functions of u1} ..., up. The function <j>(u) in the most general
* With a finite number of values.
37—2
580 NECESSARY LIMITATIONS [351
case is a generalisation of a theta function ; it will be distinguished by the
name of a Jacobian function ; but, for example, it may be a theta function,
for which, when <r < 2p, the columns a(J) are a- of the 2p columns of quasi-
periods, 2o>tf).
A consequence of the two suppositions is that in the matrix of a
columns and 2p rows, of which the (2t — l)th and 2t-th rows are formed
respectively by the real and imaginary parts of the row c^1*, ...,a,(?\ not
every determinant of a- rows and columns can vanish. For if with a arbitrary
real variables x1,...,xa. we form 2p linear functions, the (2i — l)th and
2t-th of these having for coefficients the (2i — l)th and 2*-th rows of the
matrix of cr columns and 2p rows just described, the condition that every
determinant from this matrix with a rows and columns should vanish, is
that all these 2p linear functions should be expressible as linear functions of
at most cr — 1 of them. Now it is possible to choose rational integer values
of xi} ...,0V to make all of these cr — 1 linear functions infinitesimally
small*; they cannot be made simultaneously zero since the cr columns of
periods are independent. Therefore every one of the 2p linear functions
would be infinitesimally small for the same integer values of xly...,xa.
Thus there would exist a column of infinitesimal quantities expressible in
the form ^a'1' + ••• +x<raw. Now it will be shewn to be a consequence of
the coexistence of equations (I.) that also an equation of the form (I.) exists
when a(b is replaced by an expression acja^ + ... +#aa(<r), wherein xl} ...,xa
are integers. This however is contrary to our second supposition above.
Hence also the matrix of a- columns and 2p rows, wherein the (2* — l)th
and 2i-th rows consist of a(}\ ..., a(*] and the quantities which are the
conjugate complexes of these respectively, is such that not every determinant
of cr rows and columns formed therefrom is zero.
And also, by the slightest modification of the argument, a cannot be
> 2p. The case when a is equal to 2p is of especial importance ; in fact
the case cr < 2p can be reduced to thisf case.
352. Consider now the equations (I.). We proceed to shew that in
order that they should be consistent with the condition that <f> (u) is an
uniform function, it is necessary, if a, b denote the matrices of p rows and cr
columns which occur in the scheme of § 351, that the matrix of cr rows and
columns^, expressed by
ab — ba, (A),
should be a skew symmetrical one of which each element is a rational
* Chap, ix., § 166.
+ When ff = 2p, the hypothesis of no infinitesimal periods is a consequence of the other
conditions (cf. § 345).
J The notation already used for square matrices can be extended to rectangular matrices.
See, for example, Appendix n., at the end of this volume (§ 406).
353] IN THE GENERAL DEFINITION. 581
integer. Denote it by k, so that kaa = 0, kap = — kpa. But further also we
shew that it is necessary, if x denote a column of cr quantities and x^ denote
the column whose elements are the conjugate complexes of those of x, that
for all values, other than zero, satisfying the p equations
ax = 0, (B),
the expression ikxxl should be positive. We shew that ikxxl cannot be zero
unless, beside ax, also a#j be zero : a condition only fulfilled by putting each
of the elements of x — 0 (as follows because the a- columns of periods are
independent and there are no infinitesimal periods). The condition (B) is in
general inoperative when cr < p + 1.
353. Before giving the proof it may be well to illustrate these results by shewing that
they hold for the particular case of the theta functions for which (cf. § 284, Chap. XV.)
(r = 2p, a = \ 2o>, 2o>'|, 2irib= 2rj, 2ij' ,
and therefore
ax-Za>X+^(a'X' — Qx, bx = - — .Hx,
ZTTl
where X is a column of p quantities, X' a column of p quantities, and x=
Let
X
X'
Y
y, , where, similarly, each of Y and Y' is a column of p quantities ; then*
XY' - XT=^—. (HxQy-HYQx) = ay . bx - ax . by = (ab - ba) xy=kxy,
but
where ei+wi= +1= — ti,i + p and fj,j = 0 when i~j is not equal to p ; thus we may write
kxy=XY'-X'Y=exy,
namely, the matrix k is in the case of the theta functions the matrix «•, of 2p rows and
columns, which has already been employed (Chap. XVIII., § 322).
It can be similarly shewn that in the case of theta functions of order r, k = r*.
Next if a, b, h denote the matrices occurring in the exponents of the exponential in the
theta series, we havet
namely h. ax = iriX+\>X'. Hence the equations ax = 0 give X=--.bX'. If -Xt, X±
denote the conjugate complexes of X, X' we have therefore Xl = —. \X{.
Tfl
Hence ikxx^ = ifxxl = i(XXJ - X'XJ = - 1 [bX'X^ + \ X^X'] = - - (b + bx) X'X,', since
b=b and b^b^ Thus if b = c+id, \ = c-id, the quantity -cX'X^ is positive unless
each element of X' is zero, namely, the real part of bX'X^ is negative for all values of X'
(except zero). If X'=m + in, b (m2 + »2) is equal to bm2+b?i2 ; and the condition that this
be negative is just the condition that the theta series converge.
* For the notation see Appendix n.
t Chap. x. § 190, Chap. vn. § 140.
582 PROOF OF THE NECESSITY [354
354. Passing from this case to the proof of equations (A), (B) of § 352,
we have, from equation (I.),
* [t*
_ 2Ti6(1'[« + a(» + Jo'1"] + 2iric« + 2iri&<s>[tt + £a<2)] + 27ric'2> .
where Lu = iri [6(1)a(2) - 6(2>a(1)], = - L2l. Since the left-hand side of the
equation is symmetrical in regard to ax and a2, eL" must be = eL", and
hence LK/iri is a rational integer, =k2l say, such that k12 = — k2l.
Obviously, in &u = a<1>&(3) -a(2)6(1), the part a(1'6(2) is formed by compound
ing the first column of the matrix a (of a- columns and p rows) with the
second column of the matrix 6. Similarly with a(s)6(1). Namely k12 is the
(1, 2)th element of k = ab - ba. Since similar reasoning holds for every
element, it follows that the matrix & is a skew symmetrical matrix of
integers. Conversely, if this be so, it is easy to prove by successive steps
the equation
u)
=
where
a</3
and mi, ..., mff are integers ; this equation may be represented* by
P am~| a < ^
tb?» « + -^- +27riCHi + 7ri S
L ^ J
<f>(u + am) = <f)(u)e
In fact, assuming the equation (II.) to be true for one set ml, ..., wff, we
have, by the equations (I.),
a</3
- ^am] + 27ri&(1) [M + am + Ja'11] + 2iricm + 27ri'c(1) + iri S fca)3 »ia JK^ ^ /^
a</3
_ e27ri [6m + &'1'] [M + ^aw + ^a'11] + 2^1 [cm + c'11] + vi S fca)3 7^?)^ + irf-R ^ ^ u^
For the notation see Appendix 11. — or thus —
.m1+ +b(<T)u.m<T
= bwm1.u+ +b((T)mff.u
355] OF THESE LIMITATIONS. 583
where R is equal to b(l} . am — bm . a(1), namely equal to
a</3
R + S k
1
so that
= 2 (k2lm2 + . . . + knm0) + &,2 (7^ + 1) m.2 + . . . + &1(r (m^ + l)mv + &23ra2ra3 + . . . ;
hence
iriR + iri S k-giH-iiig iri S kaoma'i>io'
6 aO = 6 a</3 ' r ,
where
[m/, . . . , wi,'] = [X + 1, 7«2, . . . , m,] ;
therefore
. r 1-1 27ri6nt'ru + inm'l + 2iricm' + 7rt S k.om'mo'i / \
<f> [u + am ] = e a</5 ap a ^ 9 (?*)•
Similarly we can take the case <f>(u + am — a(1)), noticing that equation
(I.) can be written
where v = w + ft'-".
355. The theorem (A) is thus proved. The theorem (B) is of a different
character, and may be made to depend on the fact that a one-valued
function of a single complex variable cannot remain finite for all values of
the variable.
Consider the expression
L (£) = e-^t (o+4af)-*rfcf 0 (v + a£),
wherein %l, ..., %g are real quantities.
Then £({• + «»)/£(£), wherein mlt ..., wa are rational integers, is equal
a</3
to e*ikmt+lfi s V71*7"?, as immediately follows from equation (L), and is
therefore a quantity whose modulus is unity. Now when £,..., £ff are each
between 0 and 1 and v is finite, L (£) is finite. Its modulus is therefore
finite for all real values of £ ; let 0 be an upper limit to the modulus of L (£) ;
G can be determined by considering values of £ between 0 and 1. Let now
#!, ...,xa be such that ax — 0, and let x1 denote the column of quantities
which are the conjugate complexes of the elements of the column x. Put
f = x + #1, so that af = a#j.
Then
wherein an upper limit of the modulus of L (£) is a positive quantity G whose
value may be taken large enough to be unaffected by replacing x by any
584 PROOF OF THE LIMITATIONS. [355
other solution of ax = 0 ; it is necessary in fact only to consider the modulus
of L (£) when £ is between 0 and 1.
We have
b% . a^ = b (x + x^) . a (x + x^) = bx . ax1 + bxl . axl
= bx . a#! — bxl . ax + bxl . axl = kxxl + abxf,
(c + bv) %, = w (x + xj, say, = wx + w^ + (w — w^ xlt
where w = c + bv ; therefore
gTri&f . af+27ri (c+bv) f J^ / fc\ _ giirkxx^iTrdbx^+zni (w—wj a;, gini (wx+wtx,) J^ / fc\ •
this equation is the same as
where
J£ _ £ / fc\
has the same modulus as L (£), less than G, and where
p =
— yiZj} = Zirkyz, is a real quantity (x being equal to y + iz).
Now if x be any solution of the equations ax = 0, then ^x is also a
solution, yu, being any arbitrary complex quantity and ^ its conjugate
complex. Replace x throughout by fj^x, and therefore £ by ^x + /JLXI. Then
the equation just written becomes
K having also its modulus < G.
Herein the left side, if not independent of //,, is, for definite constant
values of v and x, a one-valued continuous (analytical) function of p which is
finite for all finite values of p. Hence it must be infinite for infinite values
of ytt. Hence p must be positive, viz., values of x such that ax=0 are such
that the real quantity ikxxl is necessarily positive provided only the ex
pression
is not independent of /a.
Now if this expression be independent of p, it is equal to </> (v), the value
obtained when ft = 0, and therefore
£(0)
here the left side is a function of v provided ax± be not zero ; when ax1
is zero its value is unity ; we take these possibilities in turn :
(i) Suppose first ax^ is not zero,
355] FURTHER DEDUCTIONS. 585
then
(w — w-i) xl = (bv — b^) xl*=bx1.v — b^ . v1
must, like the left side, be a function of v and therefore a linear function, say
^-.(Bv+C), so that
<f) (v + fiax^ = <f> (v) e^H-BBM+e^ where A = iirabx? ;
hence pax^ represents a column of periods* for the function <£ (v) — and this
for arbitrary values of /A.
In this case however <f> (v) would be capable of a column of infinitesimal
periods, contrary to our hypothesis.
Hence p must be positive for values of x such that ax = 0, ax^ 4 0.
(ii) But in fact as there are a- columns of independent periods we cannot
simultaneously have ax = 0, axl = 0. For the last is equivalent to a^x = 0 ;
and ax = 0, a^x = 0, together, involve that every determinant of <r rows and
columns in the matrix
a
is zero — and thence involve the existence of
o,
infinitesimal periods (§ 351).
Hence ikxxl is necessarily positive for values of x, other than zero,
satisfying ax = 0 ; and this is the theorem (B).
Remark i. From the existence of two matrices a, b of p rows and o- columns, for
which ab — ba is a skew symmetrical matrix of integers k such that ikxxl is positive
for values of x other than zero satisfying ax = 0, can be inferred that in the matrix
, not every determinant of a- rows and columns can
ai \
vanish — and also that the o- columns of quantities which form the matrix a are inde
pendent, namely that we cannot have the p equations a,-,.r<1) + ...+ai<Tx(<r) = Q satisfied
by rational integers oX1), ..., yW. For then, also, a1^? = 0, since x=xv.
Remark ii. In the matrix k, if <r be not less than p, all determinants of 2 (o- -p) rows
and columns cannot be zero. In the matrix a, not all determinants of \<r or £ (o- + l) rows
and columns can be zero. In particular when o- = 2p, for the matrix k, the determinant is
not zero ; for the matrix a, not all determinants of p rows and columns can be zero.
Let £, TJ be columns each of or quantities. Then the coexistence of the 3 sets of
equations
is inconsistent with the conditions (A) and (B) (§ 352), except for zero values of £ and
The second of them obviously gives also at]l = 0.
For from these equations we infer that k^ = a^ . b^ - b£ . a^ is zero, and also
and therefore also k^ is zero. But by condition (B) the vanishing of ki^ when, as here,
«»7i = 0, enables us to infer rj=0.
* We use the word period for the quantities «(» occurring in our original equation (I.).
586 COMPARISON OF THE CASE [355
Similarly
is zero when I (£ + »?i) = 0, <% = (), a£ = 0. Thence by condition (B), since a|=0, £ is zero.
Suppose now that the number of the p linear functions a£ which are linearly inde
pendent is v, so that all determinants of («/ + !) rows and columns of the matrix a are zero,
but not all determinants of v rows and columns ; and that the number of the a linear
functions k£ which are linearly independent is 2**, so that in the matrix k all determinants
of 2/c + l rows and columns vanish, but not all of 2/c rows and columns. Then we can
choose 2«/ + 2K linearly independent linear functions from the 2p + o- functions «£, a^,
& (! + »?)• If this number, 2i/ + 2*, of independent functions, were less than the number 2o-
of variables |, rj, the chosen independent functions could be made to vanish simultaneously
for other than zero values of the variables, and then all the linear functions dependent on
these must also vanish.
Hence
2i/ + 2* 5 2o- or v + K > (r.
Now
i/<jt>, 2K<<r; hence v^\v, 2ic>2(o--j»).
Remark iii. It follows from (ii) that if £ = 0, then i/ = <r and <r=jo. Also that a function
of p variables which is everywhere finite, continuous and one-valued for finite values of the
variables and has no infinitesimal periods cannot be properly periodic (without exponential
factors) for more than p columns of independent periods ; in every set of o- independent
periods of such a function the determinants of a- rows and columns are not all zero. The
proof is left to the reader.
Remark iv. When a-=2p we can put a=|2&>, 2«' , wherein the square matrix 2o> is
chosen so that its determinant is not zero. When we write a—\ 2<o, 2o>' | we shall always
suppose this done.
356. Ex. i. Prove that the exponential of any quadric function of ult ..., up is a
Jacobian function of the kind here considered, for which the matrix k is zero.
Ex. ii. Prove that the product of any two or more Jacobian functions, <£, with the
same number of variables and the same value for o-, is a function of the same character,
and that the matrix k of the product is the sum of the matrices k of the separate factors.
Ex. iii. If 0 be considered as a function of other variables v than u, obtained from
them by linear equations of the form u=n + cv (p being any column of p quantities, and c
any matrix of p rows and columns), prove that the matrix k of the function $, regarded
as a function of v, is unaltered.
Obtain the transformed values of a, 6, c and bm(u+^am) + cm. (Of. Ex. i., § 190,
Chap. X.)
Ex. iv. If instead of the periods a we use a' = ag, where g is a matrix of integers with
<r rows and columns, prove that <^(u + a'm} is of the form e^ib'^('ll+Wm)+Zniem ^^ and
that V=gkg ; and also that kxy becomes changed to k'x'y' by the linear equations x=gxf,
y =gy'. In such case the form k'x'y' is said to be contained in kxy. When the relation is
reciprocal, or #2 = 1, the forms are said to be equivalent. Thus to any function $ there
corresponds a class of equivalent forms k. (Cf. Chap. XVIII., § 324, Ex. i.)
Examples iii. and iv. contain an important result which may briefly be summarised by
* That the number must be even is a known proposition, Frobenius, Crelle, LXXXH. (1877),
p. 242.
356]
OF THE THETA FUNCTIONS.
587
saying that for Jacobian functions, qua Jacobian functions, there is no theory of transfor
mation of periods such as arises for the theta functions. A transformed theta function is
a Jacobian function ; the equations of Chap. XVIII. (§ 324) are those which are necessary
in order that, for this Jacobian function, the matrix k should be the matrix e, or n
(cf. § 353).
Ex. v. If A be a matrix of 2p rows and <r columns of which the first p rows are the
rows of a and the second p rows those of b, prove that
In fact if g = Ax, g^Ax1, then
kx'x = ax . bx' - ax' .bx = 2 [& £'< + p - £/ & + p] = «££'
= f Ax . A x' = A eA . x'x.
Hence also when a- = 2p the determinant of A is the square root of the determinant of /•,
which in that case, being a skew symmetrical determinant of even order, is a perfect
square.
Ex. vi. Shew that when tr — Zp and with the notation a = |2co, 2a>'|, 2irib = \2r), 2^'|,
that
2
I 7] — T) CO, £0 ?;' — T! ft)' ,
At A = -
(a T) — tj <o, co 77 — T) a>
the notation being an abbreviated one for a matrix of 2p rows and columns. Thus in the
case when k = e, the equation of Ex. v. expresses the Weierstrass equations for the periods
(Chap. VII., § 140).
Ex. vii. In the case of the theta functions we shewed (§ 140, and p. 533) that the
relations connecting the periods could be written in two different ways, one of which was
associated with the name of Weierstrass, the other with that of Riemann. We can give a
corresponding transformation of the equations (A), (B) (§ 352) in this case, provided <r = 2jo,
the determinant of the matrix k not being zero.
As to the equation (A), writing it in the equivalent form given in Ex. v., we
immediately deduce
Ak-iA=e, (A'),
which is the transformation of equation (A).
As to the equation (B), let x be a column of a-=2p arbitrary quantities, and determine
the column 2, of <r = 2p elements, so that the 2p equations expressed by az=0, bz=x, are
satisfied. Then
thus
C(x=abz = (ab-ba)z = h, =/*, say; so that k~lu=z, k-1u,=z,
ikzzl = i (ab- ba) zzt = i (azl .bz-az.bz}) = iazl . bz = iaZjX = iaxzl = i
= ik~l Hi/j, = ik-^a^ . ax = iak ~ l a^x ;
therefore, the form
(B'),
is positive for all values of the column x, other than zero. This is the transformed form
of equations (B).
Ex. viii. When a = \ 2«, 2«' , b = l . \ 2,, 2,' | , a- = 2jo, we have
27Tl'
A(A
2o>, 2o>'
•n ni
0 - 1
•"•i
=
At -' '-N 2 / -1 -X
— 4 (coco — co co), i (COT; — co rj)
irl
1 0
K, 1
m
2 _, 1
Tl (TTI)
588
EXPRESSION OF A JACOBIAN FUNCTION.
[356
Hence when k = e, the equation (A') of Ex. vii., equivalent to A(A=-e, expresses the
Biemann equations for the periods (Chap. VIL, § 140). In the same case the equation
(B'), of Ex. vii., expresses that
= 2 2
V=l K, \ = l
is negative for all values of x other than zero.
Ex. ix. When£> = l, the two conditions (B), (B'), or
± = positive for ax—Q, iaealxlx = negative for arbitrary x,
become, for a = |2o), 2a>' , if the elements of x be denoted by x and x', and the conjugate
imaginaries by xlt x^ respectively,
i (wwj)"1 (cow/ - co'dj) x'x± = positive, i (o^w' - «&>/) xx^ = negative,
and if o>=p + io-, <»1=p-io-, a>' = p' + i<r', &>/ = p - ia-', these conditions are equivalent to
pa-' — p'<r>0,
and express that the real part of io>'/a) is negative.
357. Suppose now that cr = 2p ; we proceed (§ 359) to consider how to
express the Jacobian function. Two arithmetical results, (i) and (ii), will be
utilised, and these may be stated at once : (i) if k be a skew symmetrical
matrix whose elements are integers, with 2p rows and columns, and e have the
signification previously attached to it, it is possible to find a matrix g, of 2p
rows and columns, whose elements are integers, such that* k = geg. For
instance when p = 1, we can find a matrix such that
0 &12
-k 0
#12 #22
0 -1
1 0
ffll ff
ffa. #22 !
fu - gng* g<*g™ -
ll - #12^21 #20^12 -
namely, such that &12 = #2i#i2 — #11^22 ; f°r this we can in fact take gn, g12
arbitrarily. In general the 4p2 integers contained in g are to satisfy
p (2p — 1) conditions.
Ex. i. If a be a matrix of integers, of p rows and columns, and X be an integer, and
0, — Xa
Xa, 0
g may have either of the two following forms
ffi =
X, 0
0, a
for we immediately find p.k/ji=k.
Xa, 0
0, 1
= X, 0
0, a
a, 0
0, a-1
For a proof see Frobenius, Crelle, LXXXVI. (1879), p. 165, Crelle, LXXXVIII. (1880), p. 114.
357]
PRELIMINARY ARITHMETICAL LEMMAS.
589
Ex. ii. If /x be any matrix of integers, with 2jo rows and columns, such that ntn = f
(cf. § 322, Chap. XVIIL), we have, if k=*geff, also k=g^~l(ti-lg) and instead of g we may
take the matrix p~lff.
(ii) If g be a given matrix of integers, of 2p rows and columns, and x be
a column of 2p elements, the conditions, for ac, that the 2p elements gas
should be prescribed integers cannot always be satisfied, however the elements
of x (which are necessarily rational numerical fractions) are chosen. If for
any rational values of x, integral or not, gx be a row of integers, and we put
x = y -4- L, where y has all its elements positive (or zero) and less than unity,
and L is a row of integers (including zero), then gx = gy + gL = gy + M ,
where M is a row of integers ; in this case the row gx will be said to be con
gruent to gy for modulus g. The result to be utilised* is, that the number
of incongruent rows gx, namely, the number of integers which can be repre
sented in the form gx while each element of x is zero or positive and less than
unity, is finite. It is in fact equal to the absolute value of the determinant of
9-
For instance when c
^ is gu glz
there
are gngyz-gl
2^2! integer pair
which can be written gnx1+gl2x2, g^^ + g^,
less than unity. The reader may verify, for
the 9 ways are given (cf. p. 637, Footnote) by
vz, for (rational) values of xl} a
6 3
instance, that when q =
I 2
1
234
5
6789
T T Of
1 ? 2 "> ^
) 4 4. 5 2
¥'9 ¥» IT
i»*
2 8
f ' ¥
i t i
72285
? i?> a ¥» ¥
fi-r 4- ^T T -L 9/r ft f
v^i*/i i^ tJiX/g j «*^i i^ ^*^2 "j *-
) 2, 1 4, 1
3, 1
4, 2
5, 1 5,
2 6, 2 7, 2
To prove the statement in general let t be the number required, of integers
representable in the form gx, when x < 1. Consider how many integers
could be obtained in the form gX when X is restricted only to have all its
elements less than (a positive number) N. Corresponding to any one of the
t integers obtained in the former case we can now obtain N—l others by
increasing only one of the elements of x in turn by 1, 2, ..., N— 1. This
can be done independently for each element of x. Hence the number
of integers gX is tN" where a, here to be taken = 2p, is the number of
elements in x. Let one of these integers be called M. Then g -^ = -^r or say
M
gx = jj., wherein x is less than unity. Now when N is very great, the
p. 189
Cf. Appendix ii, § 418, and the references there given, and Frobenius, Crelle, xcvn. (1884),
590 EXPRESSION OF A JACOBIAN FUNCTION [357
M
variation of z = -^ , as M changes, approaches to that of a continuous quan
tity, and the number of its values, being the same as the number of values
of M , is
where zlt ..., za vary from zero to all values which give to x, in the equations
gas = 2, a value less than unity. Now this integral is
l....<,
0 V*l> •••! x<r)
Since this is equal to tN", it follows that t is equal to \g\, as was stated.
358. Supposing then that the matrix g, with 2p rows and columns each
consisting of integers, has been determined so that k = ab—I>a = geg, we
consider the expression of the Jacobian function when cr = 2p. The deter
minant of k not being zero, the determinant of g is not zero.
Put K=ag~l, so that K is a matrix of p rows and 2p columns, and
a = Kg ; put similarly b = Lg ; also, take a row of 2p quantities denoted by
G, such that c = gC + J [g], where c is the parameter (§ 351) of the Jacobian
function, and [g] is a row of 2p quantities of which one element is
take x, x', X, X', rows of 2p quantities such that
X = gx, X' = gx', so that ax = Kgx = KX, bx = LX, ax' = KX', bx = LX' ;
then as
kx'x, = ax . bx' — ax' . bx, = (KL — LK) X'X,
is also equal to
gegx'x = egx' . gx = eX'X,
we have
so that
i, ..., p
KxLx — Kx'Lx = (KL — LK) x'x — ex'x = S (ittaffff — xj ^J+P) '•>
i,j
further, as ikxx1 is positive for ax = 0, we have
ieXX\ = positive when KX = 0, (D) ;
jr
thus, if A denote the matrix r , we have, from the equation (C),
Li
and, if z be a row of p arbitrary quantities, and Z be a row of 2p quantities
358] BY MEANS OF THETA FUNCTIONS. 591
such that KX=0, LX = z, so that Kz = KLX = (KL - LK) X = eX, and
therefore eKz = - X, K& = eXlt we have
iKleKzz1 = positive, for arbitrary z other than zero, (F) ;
for
iKleKzzl = — iKlXzl = - i
If we now change the notation by writing K= |2o>, 2o>'|, 2mL = \2ij, 2i)'\,
and introduce the matrices a, b, h of p rows and columns defined by
a= ^r)o)~l, h = ^7rio)~l, b =
it being assumed, in accordance with Remark iv. (§ 355) that the determinant
of the matrix w is not zero, then the equation (E) shews (cf. Ex. viii., § 356)
that the matrices a, b are symmetrical, and that rf = ?7a>~1a>' — ^iriS l, so that
we can also write
r) = 2a&>, t] = 2ao>' — h', 2hw = iri, 2h&>' = b ;
also, by actual expansion,
- _ 1
b] &> = -- &>! [bj + b]
7T
WjCft), if b = c + *
7T
thus
_ 2
iKleKzzl = -- ctj, where t = wz, z and t being rows of p arbitrary quantities ;
7T
and therefore, by the equation (F), for real values of n1} ..., np other than
zero, the quadratic form bn2 has its real part essentially negative.
Hence we can define a theta function by the equation
| u . ' ) «• 2eau2+2hM(w+V
V ' — 7/ n
wherein 7, 7' are rows of p quantities given by G = (y', 7), that is, Cr = yr',
Gp+r = 7r> for r < p + 1. Denoting this function by ^ (u ; C) and taking /* for
a row of 2p integers, the function is immediately seen (§ 190, Chap. X.) to
satisfy the equation
^^^(u- C),
which is the definition equation for a Jacobian function of periods K, L and
parameter (7, for which the matrix k is e.
Further, if p be a matrix of integers with 2p rows and columns, such that
/Ze/x = e, and (Ex. ii., § 357) we replace g by fjr^g, the matrices K, L are
replaced by Kp and Z/A. Thus instead of the theta function §(u; C)
we obtain a linear transformation of this theta function (cf. § 322 Chap
XVIII.).
592 EXPRESSION OF A JACOBIAN FUNCTION [359
359. Proceeding further to obtain the expression for the general value
of the Jacobian function <£, let $ (u ; v) denote
d> (u + Kv) e~ZlriLv (U
where Vi = ni} Vi+p = ni, for i<p + l. Then, since a = Kg, and therefore
aN = KgN, we have
0 (u + aN, v) = <f) (u + KgN, v) = <j> (u + Kp, v), (I),
where /j, denotes the row gN, so that aN=Kp, N being a column of 2jt>
integers and therefore p a column of integers ; thus <£ (u + aN, v) is equal to
0 (U + aN+ Kv) e~^iLv (*+*/*+**") -2"iCv+ninn' = fy (u
where
u + K/j, + %Kv) — 2-rriCv + winn',
by the properties of (f>, N being a column of integers ; thus <f>(u + aN, v) is
equal to
<x<0
* /.. \a2TribN(u + ^aN) + 2wicN+n S kaBNaNs + 2Tri(bN .Kv - Lv .K/j.)
Q) I i-Cj l/ } c/ •
Now bN=LgN = L/j,, therefore
IN . Kv — Lv . Kp = (KL — LK} pv = epv = mri — m'n,
where ^ = mf, m+p = m/, etc. for i<p + l. If then we take v, as well as p,
to consist of integers, it will follow that
and therefore that
$ (u + aN) _ <f>(u + aN, v) _ ^ibN (u + &N) + z-wicN + ^ 2
<f> (u) (/> (u, v)
Next
(> U,
and this
= <f)(u + K/J,, v) eM,
where
M = 27riLv (u + K/J, + ^Kv) + Z-rriCv - -rrinn' - ZTTI (Lp + Lv) (u
— ZTTI (Op + Cv) + iri (m + m) (n + n') ;
therefore
<f> (U + Kfl, v) _frriLn (u
(f> (U, ft + v
iinni'+irinri—iri (m+m') (n+»i')
359] BY MEANS OF THETA FUNCTIONS. 593
of which the exponent of the right side is
iri [(KL - LK) pv - mri - m'n] = TTI \rnri - m'n - (mri + m'n)] = - Zirim'n,
so that, since /*, v consist of integers, the right side is unity.
Hence we have
(ft (U -f KfJ,, V) _ griLli (u+lKrt+tonCn-Trimm^
<ft (u, fi + v)
It is to be carefully noticed that this equation does not require /A = 0 (mod. g).
a<0
We suppose now that p=0 (mod. g). Then cN+$ 2 ka^NaN^ = Cfji,-^mm'
(mod. unity) and L(j, = bN, K^ = aN, as will be proved immediately (§ 360) ;
thus
' a
<f> (u) (f) (u, v) ~ <j>(u, p + v}
and therefore $ (u, ^ + i>) = <£ (u, v) for integer values v and any integer
values p that can be written in the form gN, for integer N '; namely (f>(u, v)
is unaltered by adding to v any set of integers congruent to zero for the
matrix modulus g.
The set of g integers gr, wherein r has all rational fractional values less
than unity will now be denoted by v, each value of v denoting a column of
2p integers — in particular r = 0 corresponds to a set of integers = p (mod. g}.
And v shall denote a special one of the sets of integers which are similarly a
representative incongruent system for the transposed matrix modulus g, such
that v =gr, the quantities r' being a set of fractions less than 1. With the
assigned values for v, let
^(w) = 2e-27rir/v0(w, v);
V
then
ijr (u + K\) = Se-2™1'" <f> (u +K\, v} = Ze-™'11 &™LK <«+***> +ar<cx-»i«' ^ (M> x + v)
for any set of integers X, as has been shewn (\ being such that, for
If now v + \ = p, so that p also describes, with v, a set of integers
incongruent in regard to modulus g, those for which the necessary fractions
s, in p = gs, are > 1 being replaced, by the theorem proved *, by others for
which the necessary fractions are < 1, so that the range of values for p is
precisely that for v, then we have
u
— g2irl'r'X+2jrt'l/X (M+iX'A)+2irtC'A— iri'M' ^g— SirirV
V
riiA. (M+JA'A) +2iriCA-irtf<' /\
* That 0(w, y) is unaltered when to »/ is added a column =0 (mod. g).
B. 38
594 EXPRESSION OF A JACOBIAN FUNCTION [359
Hence, by the result of § 284, Chap. XV., we have
1r(u) = A,*(u, C+r'),
the theta function depending on the a, b, h derived in this chapter (§ 358).
Now let v describe a set of incongruent values for the modulus g ; then
* (u, C + r')= ^ 0) = 22e-2™'" <£ (u, v) ;
v'
and since v = gr, we have v'r = gr'r = grr = vr ; thus
Vg-2jnVV _ ^ /g— 2irwV\ _ ^ (Q— 2Trirl\v\ /g— 2«r2y2 _ _ _ /g— lTrinp\v'-tp ;
v' v' v'
this sum can be evaluated :
when v = 0 (mod. g), or the numbers r are zero, its value is equal to the
number of incongruent columns for modulus g, = \g\. Since k = geg, we
have \k\ = (|#|)2, so that \g = J~k\.
when v ^0 (mod. g), so that some of rl} ..., r2p are fractional, its value is
zero, as is easy to prove (see below, § 360).
Hence we have the following fundamental equation :
Vpfcf (j> (u) = 2As* (u, C + v'\
v'
which was the expression sought.
Thus betiveen V| k \ + 1 functions <f> with the same periods and parameters
there exists a homogeneous linear relation with constant coefficients*.
Ex. i. Prove that a product of n functions ^ is a function $ for which Vl^"! is changed
into nP *J\k \. In fact the periods are na, nb.
Ex. ii. Prove that the number of homogeneous products of n factors selected from
jo + 2 functions <£ of the same periods and parameters is greater than np^\k\ when n
is large enough. And infer that there exists a homogeneous polynomial relation con
necting any p + 2, functions 0 of the same periods and parameters. (Of. Chap. XV., § 284,
Ex. v.)
360. We now prove the two results assumed.
(a) If /j, = 0 (mod. g) or ^ = gN, where N are integers, then
a<|3
cN + \ 2 k^N+Np = Cfji- %mm (mod. unity).
For
2p p
k«e = fag)*? = S (g\y(eg}yt = 2 (g)ay 2
y y=l A=l
P P P
= 2,gyaz< [ey\gw + ey,\+pg*+p,fi] + ^
y=l A=l y=
* Weierstrass, Berl. Monatsber., 1869; Frobenius, Crelle, xcvn. (1884); Picard, Poincar^,
Compt. Rendus, xcvn. (1883), p. 1284.
360] BY MEANS OF THETA FUNCTIONS. 595
therefore
a</3 p a</3
2 kaftNaN = 2 2
= 22 [ffy+p,aN..gy,itNfi+ffv,aNa.ffv+p,pNe], (mod. 2),
y-i
a</3 a</3 \
v=i
= 2 2?,gy+ptaNa . $ry,^ , (mod. 2),
y-i
where the 22 indicates that the summation extends to every pair a,
except those for which a = ft ; thus
P
= * E0*A
y-i
= 2 /iy . /iy+p = mm', (mod. 2) ;
• y=l
therefore, since %Na2 = ^Na (mod. unity), and therefore
u y=1
we have
N = {gC + ±[g]} N + ±mm' - ±\g\ N,
(mod. 1),
= gN . C + %mm' = fiC + \mm! =Cjj,- ^mm', as required.
(6) If r1} ...... , r^ be any set of rational fractions all less than unity
and not all zero and such that the row gr = v consists of integers, and
(v\, ...... , Vsp), = v', be every integer row in turn which can be represented in
the form gr' for values of / less than unity, then
is zero. Since, as remarked (§ 359), the sum can also be written
2 (e~
r'
wherein i/,, ..., v^ are integers, the sum is unaffected by the addition of any
integers to any one or more of the representants r\, .... r'ap, namely it has
the same value for all sets, v, of incongruent columns (for the modulus g).
If to each of any set of incongruent columns v we add the column
(0, ..., 0, \i, 0, ...,0), all of whose elements are zero except that occupying
the i-th place, which is an integer, we shall obtain another set of in-
congruent columns.
38—2
596
EXAMPLE OF THE EXPRESSION
[360
Suppose then in the above sum TI is fractional. Add to every one of
the incongruent sets v the column (0, 0, ..., 1, 0, ..., 0), of which every
element except the i-ih is zero. In the summation everything is unaffected
except the powers of e~2lfir^, which are multiplied by e~2nir*. Hence the
sum is unaffected when multiplied by e~Znir*, and must therefore be zero.
We put down the figures for a simple case given by
4 5
then gr=(4rl + 5rz, ?'1 + 2r2) and the equations gr = v give
thus the values of rlt r2 and vlt v2 are given by the table
ri, ra
0, 0
.i..i
1,1
"n "2
0, 0
3, 1
6, 2
Similarly gr' = (kr\+r'z, 5'''i + 2r'2), and the equations gr' = v give
thus the values of r\, r'2 and v\, i>'2 are given by the table
Thus the sum in question is
r' r'
0, 0
i, I
§, \
V'D "'2
0, 0
2, 3
3, 4
•/
3
For ?*j = Ta = v-, ••
these terms are
or zero.
For (fj, /•2) =
these terms are each unity ; for
(»"i, »-2) = (i» *), ("i» "2) = (3,
— <
3
i=l+e 3 -+e s
D> ("D "2) = (^j 2), these terms are
_2™ 2_™(2)
361. We give now an example of the expression of <£ functions.
Take the case in which p = l, and
0 -3
3 0
361] OF A JACOBIAN FUNCTION. 597
the conditions ab — ba = k, and geg = k, if a = (a, a), b = (b, b'), become
ab' -a'b = - 3, g^g* - 9u(/^ = - 3 ;
taking for instance
4 5
1 2 '
we have, if x = (x, x'), xl = (xl) &•/), and ax + ax = 0, the equation
., 9m/ ' ' \- 3tV#/ , , , __ QX'XI , ,
i/K/XJU-^ ~— Ot- ( JuJU-^ *""" CG &\) ^~ ~~^ ' \CL CL-\ "™~ CL'd-\ ) •— — ( Ct/j *~* OL yO )
tZCvj CLCt~\
where a = a. + i/3, a — a.' + ift'. Thus, beside ab' — ab = - 3, we must have
a/3' > a'ft. The quantities a, b, a, b' are otherwise arbitrary.
The equations a = Kg, b = Lg give (a, a'} = (4-K + K', oK + *2K') ; there
fore
3JT = 2a - a' , 3L = 26 - b' ,
further the equation c = gC + % [g] gives
4 1
, C") + £ (4, 10) = (46' + (7 + 2, 5C' 4- 2C" + 5),
.5 2
so that
3a=2c-c/+l, W = 4c' - 5c - 10.
Also, from K=\'2(o, 2o/ 1 , 2-iriL = , 2?;, 2?;' | , with
we obtain
a = 7rt(26-6')/(2a-a'), b = 7ri(4a'- 5a)/(2a-a'), h = 3iri/(2a - a').
If then S- (u ; (7) denote the theta function, with characteristic [ n, } ,
\-C J'
given by
^ (u ; (7) = 2eaM2+2hM(w+C)+b(«+C')!' ~2*iC' (n+C)
then the Jacobian function, with a, b as periods, and c as parameter, is given
by
where, in the three terms of the right hand, r' is in turn equal to i°
/1/3N /2/3N
\2/3j ' u/3; •
The function <f>(u) may in fact be considered as a theta function of the
third order ; its various expressions, obtainable by taking different forms for
the matrix g, are transformations of one another, in the sense of Chap. XVIII
and XX.
598 THE CASE WHEN a < 2p. [362
362. The theory of the expression of a Jacobian function which has
been given is for the case when a- = 2p. Suppose cr < *2p, and that we have
two matrices a, b, each of p rows and a- columns, such that ab — ba, = k, is a
skew symmetrical matrix of integers, for which ikxx^ is a positive form for
all values satisfying ax = 0, other than those for which also a^x = 0, or x = 0 ;
then it is possible* to determine other 2p — a columns of quantities, and
thence to construct matrices, A, B, of 2p columns (whereof the first <r
columns are those of a, b), such that AB — BA = K is a skew symmetrical
matrix of integers for which iKxxl is positive when Ax=0, except when
x= 0 or A-^x-= 0.
There will then correspond to the set A, B a function <I>, involving ^\K\
arbitrary coefficients, such that, for integral n,
The function <£ (u), which is subject only to the condition that
is then obtained by regarding <£ (u) as a particular case of <I> (u), in which
the added columns in A, B are arbitrary except that they must be such that
the necessary conditions for A, B are satisfied.
For further development the reader should consult Frobenius, Crelle,
xcvii. (1884), pp. 16, 188, and Crelle, cv. (1889), p. 35.
* Frobenius, Crelle, xcvn. (1884), p. 24.
363]
CHAPTER XX.
TRANSFORMATION OF THETA FUNCTIONS.
363. IT has been shewn in Chapter XVIII. that a theta function of the
first order, in the arguments u, with characteristic (Q, Q'), say S- (u, Q), may
be regarded as a theta function of the r-th order in the arguments w, with
characteristic (K, K'), provided certain relations, (I), (II), of § 322, p. 532, are
satisfied. Let this theta function in w be denoted by II (w, K). We confine
ourselves in this chapter, unless the contrary be stated, to the case when
(Q> Q') is a half-integer characteristic. Then the function *b(u, Q) is odd or
even ; therefore, since u = Mw, the function II (w, K} is an odd or even
function of the arguments w. Now we have shewn, in Chap. XV. (§ 287),
that every such odd, or even, theta function of order r, is expressible as a
linear function of functions of the form
•tyr(w\ K, K' + //,) = S- rw ; 2u, 2?V, 2f/r, 2£' ** + ^'r
I K J
(K' 4- LL\
-rw\ 2v, 2n/, 2£/r, 2f v ™
K
where e is + 1, according as the function is even or odd. The most important
result of the present chapter is that the functions tyr (w ; K, K' + p) which
occur can be expressed as integral polynomials of the r-th degree in 2*> theta
(n'\
w ; 2v, 2v', 2£ 2%' ) , whose characteristics are those of a
: It J
Gopel system of half-integer characteristics (Chap. XVII., § 297) ; the earlier
part (§§ 364 — 370) of the chapter is devoted to proving this theorem.
The theory is different according as r is odd or even. When r is odd,
e is ewi]Q], and we have shewn (§ 327 Chap. XVIII.) that, for odd values of r,
\Q\ = \K\, (mod. 2) ; the theory deals then only with functions
*
\lrr (w ; K, K' + //.)
600 GENERAL STATEMENT. [363
in which e = elrfl-K'1. When r is even, e, though still equal to e77*^1, may or
may not be equal to e7™1 Kl, according to the integer matrix which determines
the transformation ; but in this case, also, the value of e in the functions
tyr(iu; K, K' + /LI) which occur is determinate.
The proof of the theorem is furnished by obtaining actual expressions for
the functions ^rr (w ; K, K' + /*) as integral polynomials of the r-th degree in
/ R'\
the 2? functions $• I w ; 2v, 2t/, 2£ 2£" j ' ) ; the coefficients arising in these
\ | -R /
polynomials are theta functions whose arguments are r-th parts of periods,
of the form (2vm + 2i/w')/r. The completion of the theory of the trans
formation requires that these coefficients should be expressed in terms of
constants depending on theta functions with half integer characteristics
(§ 373).
Further the theory requires that the coefficients in the expression of the
function II (w ; K} by the functions ^,, (w ; K, K' + /A) should be assigned
in general. In simple cases this is often an easy matter. The general case
is reduced to simpler cases by regarding the general transformation of the r-th
order as arising from certain standard transformations for which there is no
difficulty as to the coefficients, by the juxtaposition of linear transformations
(|| 371—2)*.
364. It follows from § 332, Chap. XVIII. that any transformation may
be obtained by composition of transformations for which the order r is a
prime number. It is therefore sufficient theoretically to consider the two
cases when r = 2, and when r is an odd prime number. We begin with the
former case, and shew that the transformed theta function can be expressed
as a quadric polynomial in 2^ theta functions belonging to a special Gb'pel
system. A more general expression is given later (§ 370).
* For the transformation of theta functions, and of Abelian functions, the following may be
consulted. Jacobi, Crelle, vm. (1832), p. 416 ; Eichelot, Crelle, xn. (1834), p. 181, and Crelle,
xvi. (1837), p. 221 ; Eosenhain, Crelle, XL. (1850), p. 338, and Mem. par divers Savants, t. xi.
(1851), pp. 396, 402; Hermite, Liouville, Ser. 2, t. in. (1858), p. 26, and Comptes Rendus, t. XL.
(1855); Konigsberger, Crelle, LXIV. (1865), p. 17, Crelle, LXV. (1866), p. 335, Crelle, LXVII. (1867),
p. 58; Weber, Crelle, LXXIV. (1872), p. 69, and Annali di Mat. Ser. 2, t. ix. (1878); Thomae,
Ztschr. f. Math. u. Phys., t. xn. (1867), and Crelle, LXXV. (1872), p. 224 ; Kronecker, Berlin.
Monatsber., 1880, pp. 686, 854 ; H. J. S. Smith, Report on the Theory of Numbers, British Associa
tion Reports, 1865, Part vi., § 125 (cf. Weber, Acta Math., vi. (1885), p. 342; Weber, Elliptische
Functionen (1891), p. 103; Dirichlet, in Riemann's WerJce (1876), p. 438; Cauchy, Liouville, v.
(1841), and Exer. de Math., n., p. 118; Gauss, Werke (1863), t. n., p. 11 (1808), etc.; Kronecker,
Berlin. Sitzungsber. 1883 ; Frobenius, Crelle, LXXXIX. (1880), p. 40, Crelle, xcvn. (1884), pp. 16,
188, Crelle, cv. (1889), p. 35 ; Wiltheiss, Crelle, xcvi. (1884), p. 21 ; the books of Krause, Die
Transformation der Hyperelliptischen Functionen (1886), (and the bibliography there given),
Theorie der Doppeltperiodischen Functionen (1895) ; Prym u. Krazer, Neue Grundlagen einer
Theorie der allgemeinen Thetafunctionen (1892), Zweiter Teil. See also references given in
Chap. XXI., of the present volume, and in Appendix n.
*C\BRT^
OF TTTF
UNIVERSITY
364] TRANSFORMATION OF THE SECOND ORDER. 601
By means of the equations u = Mw, a function S- fn ; 2w, 2<w', 2?;, 2rj' } ,
with half-integer characteristic ( ] , becomes a theta function in lu,
\Q I
n (w ; K, K'}, of order 2, with the associated constants 2u, 2i/, 2£ 2^' and
the characteristic (K, K'), where (§ 324, Chap. XVIII.)
• 2o//3',
= && a/3' - a'yS = /8'a - y8a' = 2 ;
and
this theta function in w, U(w; K, K'), can by § 287, p. 463, be expressed as
a linear aggregate of terms of the form
; 2u, 2n/, 2
- rw
', 2(r/r,
r being equal to 2 ; here e, = eMQQ' , is + 1, according as the original function,
that is, according as the function II (w ; K, K'}, is even or odd. For brevity
we put w = 2vW, VT' = V, and denoting by @ (W, r) the series
we consider the function
.8
[-rF; »V (A"+f/r],
which is equal to e-Mv~lvfltyr(w ; -ff", K' + p}. Throughout the chapter the
/ K'\ f I ^'\
symbols ^ ( w } , ® (W • } denote respectively
\ K. / \ I K /
; 2u,
/o'\ /r'\
Taking the final formula of § 291, p. 472, replacing co, •', 1^, 9, ( ), I J
respectively by v, v, f, 5", -|(a ) , %{ } + ( K } , multiplying both sides of the
\ct j \a / \ K I
equation by e«a(M-tf'-<O; where ft is a row of integers each either 0 or 1, and
adding the %' equations obtainable by giving a all values in which each of its
elements is 0 or 1, we obtain
602
EXPRESSION OF THE TRANSFORMED FUNCTION
[864
V
•IT'
' + *')
K
and hence, replacing V, U respectively by W, 0,
2F";
i-*'}
This may be regarded as the fundamental equation for quadric transformation ;
we consider various cases of it.
(i) When (K, K') is the zero characteristic we obtain
; 2r'
:
the right-hand side being independent of a', which for simplicity may be
put = 0.
We can infer that in any quadric transformation, when the transformed
function has zero characteristic, it can be expressed as a linear aggregate of the
, in which «' is an arbitrary row of integers (each 0
z^ squares 5r I w
or 1) and a has all possible values in which its elements are either 0 or 1.
(ii) When JK' = Q, K=^n is not zero, we obtain
where on the right side only 2^-1 terms are to be taken in the summation in
regard to a, two values of a whose difference is a row of elements congruent
(mod. 2) to the elements of n not being both admitted. When jH ) is an
even characteristic we may put a' = 0 ; when Jj is an odd characteristic we
may put a' = /i.
In this case, as before, only 2^ theta functions enter on the right hand,
and their characteristics form a special Gopel system.
The cases (i) and (ii) give the transformation of any theta function when
the matrix, of 2p rows and columns, associated with the transformation* is
For the notation, cf. Chap. XVIII., §§ 322, 324..
365] BY MEANS OF A CERTAIN GOPEL SYSTEM. 603
/20
( . It can be shewn that by adjunction of linear transformations every
quadric transformation is reducible to this case (cf. § 415 below); so that
theoretically no further formulae are required. As it may often be a matter
of difficulty to obtain the linear transformations necessary to reduce any given
quadric transformation to this one, it is proper to give the formulae for the
functions
2r' *"^1 +«6 F-2TT 2r'
by this means the problem is reduced to finding the coefficients in the
expression of any theta function in w, of the second order, in terms of
functions W2(W; K, K' + yu.) (see § 372 below). Hence we add the following
case.
(iii) When K' is not zero, we deduce, by changing the sign of W in the
fundamental formula, the equation
;
2r' 2(JF; K,
;
where, putting K= |&, K' = ^lc, we have Ca = 1 + eeniklk'+a'}+niak' . When e is
+ 1, there are 2f~l values of a for which atf = k(k' + a') + 1 (§ 295, Chap. XVII.);
for these values Ga = 0 ; when e = -l, there are 2?-1 values of a for which
ok' = k (kr + a') ; for these values Ca = 0. In either case it follows that the
right side of the equation contains only 2?~l terms, and contains only 2^
theta functions whose characteristics are a special Gopel system.
It is easy to see that the results of cases (ii) and (iii) can be summarised
as follows: when the characteristic (K, K') is not zero the transformed function
is a linear aggregate of 2*-1 products of the form ^ [w ; A, P{] ^ [w ; A, K, P,:]
wherein the 2"-1 characteristics P{ are of the form | fj , K=(K }, and
\a/ V— K)
A, K are such that* e"*i *i +»* i A *i = e.
These results are in accordance with § 288, Chap. XV. ; there being
2P-i (1 + e) linearly independent theta functions of the second order with
zero characteristic and of character e, namely 2*> such even functions and no
odd functions, and there being 2?-1 linearly independent theta functions of
the second order with characteristic other than zero.
365. Ex. i. When p = l, the results of case (i), if we put egh(W; r') for
9 I w > T' k ( _ f ) I , as is usual, are
800(2 w- yj-^^^+oiiWO eL(jV
2000(20 -ae
* For the notation, see Chap. XVII., § 294.
604 TRANSFORMATION OF THE SECOND ORDER [365
and
where e (2r') denotes e (0 ; 2r'). If then we introduce the notations
VI e^r') e01(2r') /X_e10(r') /r>_e01(r')
' "'' VX-~ VX-
4 v_
; 2r')' ~ V * 001 (2 TF; 2r') ' * ~ 001 (2 W- 2r')'
r')
; r')'
= -1 ^liJ^JL) ./- A A! Qio(JF;j-') i/?-^6"^ r')
VAe01(JF; r')' V>?- V X 601(Pf; r')' Vf~VX e
we find by multiplying the equations above that
and therefore that
so that also
while, comparing the two forms for 0^ (2 IF; 2r'), putting 1F=0, we obtain
/ / X , 1 — X' . . 2 V^
giving K*
further the equations for 600(2 W; 2/) and 010(2TF; 2r') give the results
from which we find
r? = l-£, C=1-X^; thus also // = 1-A-, z = I-k'*x.
Ex. ii. The equations of case (ii), also for p = l, give
eol(2if; ^^(^O^^O 20=e10(ir;.Qe
yoi V^r -> 60i (2r )
From these we have by division
while from these and the results of Ex. 1, we find
]/ Vi^X^, \/~z= [i - (i - x') ^l/Vr^W-
t-. iii. When p = 1, by considering the change in the value of the function
3* (w] — P11
^
when ?y is increased by a period, we immediately find that it is a theta function in w of
the second order with characteristic £ (j • hence by the result of case (iii) above, the
function is a constant multiple of 310 (w) Sw (w) ; determining the constant by putting
iv =0, we obtain the equation
©oo(Oe10(r')[0'u(TF; r')001(JF; r')-e'ol(JF; r)0n(IF; r')]
= e'u (r') 001 (r') 610 ( W', r') 0oo ( W; r'),
365] IN THE ELLIPTIC CASE.
which is immediately seen to be equivalent to
e'u(r') 600(1-') w= ft d£
©01 (O e10 (r') J 0
605
[We may obtain the theta relation, here deduced, from the addition formula of Ex. i.,
§ 286, p. 457 ; taking therein m = $( ^\ , a^=\ ( _ A , «2 = | ( _ i) ' W=G> ? = ^ (Q) '
r=p = % ( ) , we immediately derive
^10 (w) Soo («) 3U (2») V (0) = 5oo («0 310 («0 [$oi 0* - *>)
if, for small values of v, this equation be expanded in powers of v, and the coefficients of v
on the two sides be put equal, there results the equation in question.]
Ex. iv. By differentiating the second result of Ex. ii., putting TP=0, and putting
TF=0 in the first result of the same example and in the second value for 600(2 W; 2i-') in
Ex. i., we obtain
600 (2r') 601 (2r') 610 (2r') 6^ (r') 601 (r') 610 (r) '
so that the second of these functions is unaltered by replacing T by 2'V, n being as large
as we please. Hence we immediately find from the series for the functions, by putting
T = oc , that each of these fractions is equal to TT. Hence if the integral occurring in the
last example be denoted by J we have ,7=776^ (r) W. In precisely the same way we find
7=2776^(21-') W, where 7 is an integral differing only from J by the substitution of x for $
and k for X. Hence
as follows from the first result of Ex. 1.
From these results we are justified in writing the formula of Ex. ii. in the form
' T+X'~ dn(/,X)
and this is Landen's first transformation for Elliptic functions.
Ex. v. The preceding examples deal, in the case p = 1, with the quadric transformation
associated with the matrix ( j . Prove when p = 1 that for any matrix of quadric
transformation the transformed theta function is expressible linearly in terms of one or
more of the eight functions
6 =
; 2r'),
2 = 610(2TF; 2r'),
6 (W; 2r'
= 601 (2 W; 2r'),
j 2r'),
4)- 6 (2 IF; 2r'
-^2 TF;
67 = 6 2 TT;
Prove in particular that the functions arising for the transformation associated with
the matrix ( ) are expressed as follows :
\U ^/
, e,0(Tr; ^-')=e4, en(Tf; £i-')=-tea;
606 EXAMPLE OF THE ELLIPTIC CASE. [365
and that the functions arising for the transformation associated with the matrix ( ) are
\U 2J
expressed as follows :
©oo(^; £r'-i) = e-ie2, e01(TF; £r'-|)
_
ew(W;br'-to = e ~^ ee, eu( W; £r'-|) = e8 er.
Obtain from the formulae of the text the expressions of the functions 64, 05, 86, 07 of
the form
), e5=6'5e01(TF)en(TF), ec=<70e01(TF)e10(TF), er=C'7
where 6'4, C5, Ce, C"r are constants.
Ex. vi. The reason why the matrices (A1), (A0), (no) are selected in Ex. v. will
\u i/ \u // \u zj
appear subsequently (§ 415) ; the matrix ( ) gives the transformation which is supple-
U /
mentary to that given by L J ; it gives results leading to the equation
sn [(1 +k) u, 2
by combination of these results with those for the matrix ( j we obtain the multiplica
tion formula
0n(2TF; r') = ^en(TF; r')e01(Pf; r')e10(^; r')6oo(TF; r'),
where ^ is a constant (cf. Ex. vii., § 317, Chap. XVII. and § 332, Chap. XVIIL).
The matrix associated with any quadric transformation can be put into the form
where fl, Q' are matrices of linear transformations ; for instance we have
0 - 1\ /2 0\ / 0 1\ _ /I 0
Oj (o l) V-l 0/~ \02
with the corresponding equations
U=rW^ TF1 = 2TF2, >F2=-r2PF3; r^-l/r, r2 = r1/2, rs=-l/r2,
from which we have, for instance,
- 901 (2 TF2 ; 2r2)
= e~ - EQm( TF2 ; r2) G01 ( T<F2 ; r2) = ^e^ ( TF3 ; r3) 010 ( TF3 ; r3),
(E, F being constants) whereby the transformation formula for 010 ( TF3 ; ^r3) is obtained
from those for 610(2 TF; 2r'), with the help of those arising for linear transformation.
366. We pass now to the case when the order of transformation is any
odd number, dealing with the matter in a general way. Simplifications that
can theoretically be always introduced by means of linear transformations are
considered later (§ 372).
360] TRANSFORMATION OF ANY ODD ORDER. 607
We first investigate a general formula* whereby the function
can be expressed in terms of products of functions with associated constants
2v, 2t/, 2£, 2£'. We shall then afterwards employ the formulae developed in
Chap. XVII., to express these products in the required form.
Let a, a-' be two matrices each of p rows and m columns, whose constitu
ents are any constants ; let the j-th columns of these be denoted respectively
by <r® and <r'&, so that the values of j are 1, 2, ..., m; let T, denote the
matrix 2v<7 + 2i/<r', which hasp rows and m columns, and let the j-th column
J
of this matrix, which is given by 2i;o-(^ + 2wV^, be denoted by TVJ ; also,
K, K' being rows of any p real rational elements, let TK, ZK denote the
rows 2vK+ 2v'K', 2£K+2£'K' ; and use the abbreviation
«• (w ; K, K') = ZK (w + %TK) - TriKK' ;
finally, let s = (s(1), ..., s(w)) be a column of m integers whose squares have
the sum r, so that
then, using always ^ (w) for ^ (w ; 2v, 2i/, 2^, 2^)> the function
Uw = e~ rw lw ' K^ K'W 3 ^
is, m w, a theta function of order r with associated constants 2u, 2u', 2f, 2£'
characteristic (K, K'}.
For when the arguments w are increased by the elements of the row
where N, N' are rows of p integers, the function
is multiplied by a factor e*i, where tyj is equal to
that is
the sum of the m values of ^- is given by
7 = 1
* Konigsberger, Crelle, LXIV. (1865), p. 28. See Eosenhain, Crelle, XL. (1850), p. 338, and
M£m. par divers Savants, t. xi. (1851), p. 402.
608 THE FIRST STEP IN THE [366
also, when w is increased by Ty, the function — rtx [w ; K/r, K'\r) is increased
by — ZjrTA-; thus the complete resulting factor of II (w) is
of which (§ 190, p. 285) the exponent is equal to
m (w ; N, N'} + Ziri (NK' - N'K) ;
thus (§ 284, p. 448) II (w) is a theta function in w, of the r-th order with
(K, K') as characteristic.
Therefore (§ 284, p. 452) we have an equation
II (w) = 2 A^ \rw ; 2v, 2rv', 2£/r, 2£' ^' +
n L K
where yu, is a row of p integers each positive (including zero) and less than r,
and the coefficients A^ are independent of w. The coefficients A^ are inde
pendent of K, K', as we see immediately by first proving the equation which
arises from this equation by putting K and K' zero, and then, in that equation,
replacing w by w + ZvKjr -f- 2v'K'/r.
In this equation, replace K by K + h, where h is a row of p integers, each
positive (including zero) and less than r ; then, using the equation previously
written (§ 190, p. 286), for integral M, in the form
*(u; q + M} = e^iM(i^(u\ q\
we find
- rat [w ; (K+ h)jr, K'/r] - 2« (K' + e) ft/r g ^ (})
j=i
rw ; 2v,
K
where e is taken to be any row of p integers each positive (or zero) and less
than r ; ascribing now to h all the possible r? values, and using the fact that
h
according as /* — e = 0 or ^ 0, (mod. r), we infer, by addition, the equation
C^ rw ; 2v, 2n/,
v> + - I + TJ
h j=i |_ \ r
where
•^r = — TOT [w ; {K + h)/r, K'/r] — 2jri (K' + /A) hfr,
and C^, = rPAp, is independent of w and of the characteristic (K, K').
367] TRANSFORMATION OF ODD ORDER. 609
367. We put down now two cases of this very general formula : —
(a) if each of the matrices <r, a-' consist of zeros, and each of the m
integers s(1), ..., s(m) be unity, so that m = r, we obtain
, 2n/, 2£/r, 2£'
r' K '^ ~ ir r w + -
r
In using this equation we shall make the simplification which arises by
putting w = 2u W, v~lv = r, and
@ ( W, r) = e-M-w * (w) = 2e*riwr»+*T'»8;
71
then the equation can be transformed without loss of generality, by means of
the relations connecting the matrices v, v, f, £' (cf. § 284, p. 447), to the form
(I)
(7 e-<twiK"\_W+WK'lr-\-lTnKK'lr® rW, TT
where Cy is independent of W and of K and K'.
This equation is of frequent application in this chapter ; it is of a different
character from the multiplication formula given Chap. XVII., § 317, Ex. vii.,
whereby the function © (rW, r') was expressed by functions @(W, r') with
different characteristics but the same period, r'.
Ex. \. When r = 2, p = 2, we have
F, 2r')=e2(Tru wz- r'j+e^i^+i, TF2; r')+ e2(iF15 ^2+i; r')
Ex. ii. If X, /LI, /i be rows of p integers each less than r, prove that the ratio
2<r*n^/rer T |p+ J X/rH ^^^.p^ T Tf+ AJ
is independent of W.
(/3) if the matrix or' consist of zeros, and if each of the m integers
sw, ..., s(m) be unity, so that m = r, and if the matrix <r, of p rows and r
columns, have, for the constituents of every one of its rows, the elements
i ?
,
r r
then the matrix Tff will have, for the constituents of its i-ih row, the
elements
H. 39
610 THE FIRST STEP. [367
where fl; is the sum of the elements of the i-ih row of the matrix 2i>,
so that
p
fit = 2 2 vi h ;
A = l
also the *-th of the p elements denoted by - T^s will be
(r-i)nri_r-i
~
and therefore the i-th of the elements of T^ -- T0s will be
r
r 2r
Thus, denoting the row (fllf ..., flp) by ft, the theorem is
A ,-=i
where -^ has the same value as in § 366. And as before this result can be
written without loss of generality in the form
[rW, rr' /
where U '= W — (r — l)/2r and, for any value of u,
<j)(u) = ®(u; r) 0 ( u + - ; r'j ...... 6 f w + 7 - ; T' J ;
the number of different terms on the right side of this equation is r?~l ;
for if m be a positive integer less than r, the two values of h expressed by
h = (hrl> ..., hp) and h = (h1', ..., hp'\ in which h^li^ + m, ..., hp' = hp+m,
s i \ • ±1, ^ f ^fTT,h + ]K: + r^\
(mod. r), give the same value tor <p I u H -- • I .
Ex. i. For j» = 2, r=2, we obtain
')e(F1+J, JT2 + i; r')
+i, TF2-i; r'JeC^-i, TT2 + |; r').
r. ii. For jo = 2, r=3, we obtain, omitting the period r on the right side,
r') = e(TT1, ^0(1^-1, Tf2-J)9 (W,+^, TT2+i)
368]
THE SECOND STEP.
611
368. We consider now the expression of the function
%(r; K,K- + rt-»[rW; rr' (A" + ")/r] + e© \-rW ; rr' j <""' + *>
[• jR'~l
W ; T' i , in the case when ?* is odd. We
suppose as before (K, K') to be a half-integer characteristic, and we suppose
e = e*i\K\^ so ^hat e is + i according as the characteristic (K, K') is even or
odd*. It follows from § 327, Chap. XVIII., if (K, K') has arisen by trans
formation of order r from a characteristic (Q, Q'), that e is also equal to ewi ' ^ '
and is + 1 according as the function is even or odd.
It is immediately seen that equation (I) (§ 367) can be put into the form
h r J
from this equation by changing the sign of W, we deduce the result
where we have replaced ee~^irKK' , = ee-™-\K\ by unity, and a denotes the
expression [h -(r-l) (K + r'K')]/r, which is an r-th part of a period. We
proceed to shew that the function
K
i (r-l) K'WQr
can be expressed as an integral polynomial of the r-tli degree in 2^ functions
©'•[I'F; T' APi], where APt are the characteristics of any Oopel system of
half-integer characteristics whereof (K, K') is one characteristic.
From the formula of § 311, p. 513, putting C=0, A' = A,B = P =%(q'} ,
fP \ / P \
and replacing U, V, W, fVJ €i, (jM e; respectively by Tf, a, b, €i, e,- we
obtain, if Pa =
* Thus, when 2(K' + /j.)-rm, m being integral,
e_e2iriK(rm-2p)_ ZwiKm_
as in § 287, Chap. XV., and
tyr(W; K, K' + u.) reduces to 29
K'+u.
39—2
612 THE SECOND STEP IN THE [368
; A)
; A + P.),
> ) *> o.
where
X (u, v ; P, e) = 2e0e-W<?a 0 (M ; A + P + Pa) © [> ; 4 + Pa] ;
the function x(u> v> A, P, e) may be immediately shewn to be unaltered by
the addition of an integral characteristic to the characteristic P0 of one of its
terms ; we may therefore suppose all these characteristics to be reduced
characteristics, each element being 0 or |.
Hence we get
; A+Pa)®(W; A + P.),
and hence 2^®3(TF+a; A) is equal to
^H^ea@(W- ^+Pa)2#22e/e-^'V?/3@(TF+3a; A + P,L+Pft)®(W;A+Pp),
e a. e' |3
where
77 = x (g» g ; °> 6) 77 = y (2a> a ; -Pa. 6/) .
%(2a,0;0,e)' X (3a, 0 ; P., e')'
proceeding in this way we obtain 2(r~1)i} ©r ( W+ a ; J.)
2; CM>, (ni)
where each of P0i, P02, ... becomes in turn all the characteristics of the
group (P), and e1} e2, ... relate respectively to the groups described by
Pa,, Pa.,, •••, and further
+ l)a, 0; P0i + ... + P.^, em],
(m=l, ..., r-1),
em = eay™@ ( If ; ^1 + P. J, X^ = - ^TTI (?'0l + ... + j'a^^) ?am,
(m=l, ...,r-2).
The equation (III) expresses ®r(W + a; A) as an integral polynomial
which is of the (r - l)th degree in functions © ( W ; J. + Pa), whose charac
teristics belong to the Gopel system (AP), arid is of the first degree in
functions ® [W + ra; A +P0]. But it does not thence follow when a is an
r-th part of a period, that ®r(W + a; A) can be expressed as an integral
polynomial of the r-th degree in functions @[TT; .4+P0]; for instance
if the Gopel system be taken to be one of which all the characteristics are
even (§ 299, Chap. XVII), it is not the case that the function @3 (W + £),
368] TRANSFORM ATION OF ODD ORDER. 613
which is neither odd nor even, or the function ®3(W+ ^) — Ba( W— £), which
is odd, can be expressed as an integral polynomial of the third degree in the
functions of this Gopel system ; differential coefficients of these functions
will enter into the expression. The reason is found in the fact noticed in
§ 308, p. 510 ; the denominator of Hr^ may vanish.
Noticing however, when P is any characteristic of the Gopel group
(P), that x(-u,-v; P, e) = e«i^i+«i^-Pi x (u, v; P, e), so that the co
efficients Hm are unaltered by change of the sign of a, and putting the
(K'\
„ J , we infer, from the equation (III), that
is equal to
2^2 ...... 2#r_1|>-2™<»--1>*'^%(TF + ra, W; P, eM)
e, ai «r_i
+ e***W'irx(W-ra, W ; P, e^)],
where P denotes Pfli + . . . 4- Par 2 ; and it can be shewn that when a becomes
equal to [h -(r-l)(K + r'K')]/r, the limit of the expression
U=Hr_1 [e-^(r-vK'wx (W+ra, W; P, er_0 + e^^K'wX (W-ra, a; P, er_0],
if it is not a quadratic polynomial in functions @(TT; APa), is zero. The
consequence of this will be that ^r [ W ; K, K' + //,] is expressible as a
polynomial involving only the functions ® (TF; APa).
For the fundamental formula of § 309, p. 510, immediately gives*, for
any values of a, b,
x(W+a, W + b; P, e)x(a + b, 0; P, e) = x(a, b; P, €)x(W + a + b, W; P, e),
and hence, replacing e,._j simply by e, the expression U is equal to
2eae-*"V9. {e-'-o-D K>W® (W + a; A +PJ@[W + (r-l)a; A + P + PJ
where P, =i> is used for Pa+ ...... + P and e1; e2,... for
(er-i)ai •••• Replacing ra in this expression by the period h-(r-l)(K+r'K'),
and omitting an exponential factor depending only on r, h, K, K' and P, it
becomes
@[W-a; A + P + Pa]
* We take the case when the characteristics B, A of % 309 are equal. It is immediately
obvious from the equation here given that in the expressions here denoted by Hm the value of the
half-integer characteristic A is immaterial.
614 FORMULATION OF THE GENERAL THEOREM [368
A being as before taken = ( ^J and £ = «.tf^»> W«^fr-*>*'«» ; and this
is immediately shewn to be the same as
*\ A+Pa)S(W-a; A + P + Ptt),
where ep is the fourth root of unity associated with the characteristic P of
the Gb'pel group (P), which is to be taken equal to 1 in case P = 0. Thus
/A\
the expression vanishes when fp = — e*ni ' p ' ( „ J . Hence, in order to prove
that when the expression U is not a quadratic polynomial in functions
© (W ; APa), it is zero, it is sufficient to prove that the only case in which
fA\
U is not such a quadratic polynomial is when £P = — e?ni ' p ' ( p ) •
Now the denominator of Hr^ is
2e0e-^9'3« 0 [ra ; A + P + PJ @ [0 ; A + Pa],
a
where P still denotes Pai + . . . + Pa and ea has the set of values of er_2 ;
save for a non-vanishing exponential factor this is equal to
or
according as P = 0 or not, where, in the second form, P^ is to describe a
group of %P~I characteristics such that the combination of this group with
the group (0, P) gives the Gb'pel group (P). We shall assume that, when
fA\
£P is not equal to — e*wi ' p ' ( p j , neither of these expressions vanishes for
general values of the periods r.
Since the function *¥r ( W ; K, K' + /jJ) is certainly finite, we do not
examine the finiteness of the coefficients Hm when m is less than r — 1,
these coefficients being independent of W ; further, in a Gopel system (AP),
any one of the characteristics APa may be taken as the characteristic A ;
the change being only equivalent to adding the characteristic P0 to each
characteristic of the group (P); hence (§ 327, Chap. XVIII.), our investigation
gives the following result : — Let any 2? functions ^f(u', 2o>, 2o>', 2??, 277'
whose (half-integer) characteristics form a Gopel system, syzygetic in threes, be
transformed by any transformation of odd order; let (AP) be the Gopel
system formed by the transformed characteristics [ j J then every one of the
369] FOR TRANSFORMATION OF ODD ORDER. 615
original functions is an integral polynomial of order r in the %> functions*
*b(w ; 2u, 2i/, 2£ 2f | AP) : as follows from § 288, Chap. XV., the number of
terms in the polynomial is at most, and in general, \ (/* + 1).
For the cases p = l,2, 3, and for any hyperelliptic case, it is not necessary
to use the addition formula developed in Chap. XVIII. We may use instead
the addition formula of § 286, Chap. XV. It is however then further to be
shewn that only 2? theta functions enter in the final formula. For the case
p=3 the reader may consult Weber, Ann. d. Mat. 2a Ser., t. IX. (1878),
p. 126.
369. We give an example of the application of the method here followed.
Suppose p = l, r = 3, and that the transformation is that associated with the matrix
; then (§ 324, Chap. XVIII.) taking l/=3, the function
9[u; 2», 2o>', 2,,, 2,/ | !(_?)],
or SOI(M), is equal to 501(3zc; 2v, 6v', 2f/3, 2f) or ^r)<1>"1">3( W; -£, 0). Now we have,
also e^j ( W+ a) is equal to
a a,
a, 0; 0, e) a e/x(3«, 0; ^ a, e ) ^ »
if we take the Gopel system to be \ ( j , £ L J , so that *V"i(i)j tnis is e(lual to
e^(a)+fle;o(q) e01 (2a) e01 (a) +«1'e10 (2a) e10 (a)
J,e01(2a)e01+fle10(2a)e10 ' > t, e01(3a)e01+fl'e10(3«)e10
6l(> (2a) 9()1 (a) " *' 1/e<)1 (2a) QI° (a) '
,!„ oi—
"I:e01(2a)e01+«1e10(2a)e10€l e, e10(3a)e01-iVe01(3a)e,0
where 001 denotes 001 (0), etc., and
iV1'801( ir+3a.)010( IV).
Now, in accordance with the general rules, the denominator of the fraction
610 (2a) 001 (a) - if/e01 (2a) 010 (a)
e10 (3a) e01 - ifl' e01 (3a) e10
vanishes when ./= -^ e-^-2A')?l'+«2^s namelyj as =i _ = ^, when
,,'= _le«^+1), and a = (A + l)/3 ; in fact, putting o =
e10 (3a) e01 - W/GO! (3a) e10= e'r£(t+1) e10 (.r) e01 - iV
* The expression of the transformed theta function in terms of 2? = 4 theta functions is given
by Hermite, Compt. Eendu*, t. XL. (1855), for the case p = 2. For the general hyperelliptic case
cf. Konigsberger, Crelle, LXIV. (1865), p. 32.
616 EXAMPLE OF THE GENERAL METHOD FOR ODD ORDER. [869
for small values of .r, when iel' = e1rl^ + ', because the differential coefficients of the even
functions, being odd functions, vanish for zero argument ; thus the denominator of the
fraction vanishes to the second order. We find similarly, for ie1' =
that the numerator of this fraction is equal to
in the same case also we find that the expression El is equal to
e« <*+1> [e'10 ( W) e01 ( W) - e'01 ( W) e10
while the expression 010 ( W- 3a) 001 ( W) — ?'e/e01 ( W- 3a) 010 ( W) is equal to the negative
of this. Thus the function ©^ ( W+a) can be expressed by the functions 010 ( TF), 001 ( W),
and their differential coefficients of the first order ; but the function ©^ ( W+a) + 0^ ( W— a)
can be expressed by the functions 010 ( W), Qol ( W) only.
In the function 0^ ( W +a) + 0^ ( W- a) the part
s 010 (2a) 001 (a) - ^/001 (2cQ 010 (a) £
e/ 010 (3a) ©01 - tV ©01 (3a) 010
furnishes only the single term for which ie{= -em (7t+1), namely,
h-±l\
T •*
GOI ( W) 010 ( TF).
e -e
*% \ o I *T
\ * /
U01 U10
Ex. i. Prove that the final result is that ?C0301 (u) is equal to
- [ej, (i) e^ - ©^ (i) ©J0] »a (w)
(j) [©10 (j) ©01 + ©01 (j) 010] n 2 3
where ©01, ©10 denote ©01 (0) and ©10 (0) respectively.
Ex. ii. Prove that
©in (i) ©n
=2-—
370. General formulae for the quadric transformation are also obtainable.
The results are different, as has been seen, according as the characteristic
(K, K') of the transformed function is zero (including integral) or not. The
results are as follows : —
When (K, K') is zero, the transformed function can be expressed as a
linear aggregate of the 2^ functions S-2 (w A, Pi), whose characteristics are
those of any Gb'pel system,
370] GENERAL THEOREM FOR TRANSFORMATION OF THE SECOND ORDER. 617
When (K, K') is not zero, the transformed function can be expressed as a
linear aggregate of the 2*"1 products ^ (w \ A, PI) ^ (w \ A, K, Pi), in which
the characteristics Pi are those of any Gopel group whereof the charac
teristic K, = (K, K'), is one constituent, and A is a characteristic such that
| A, K | = | K |, or | A, K \ = \ K \ + 1 (mod. 2), according as the function to be
expressed is even or odd*.
When (K, K') is zero, the equation (I), § 367, putting K = K' = /* = 0,
and then increasing W by ^/u-r', where /A is a row of quantities each either
0 or 1, gives
2r'
o )
0
hence, from the fundamental formula of § 309 (p. 510), writing therein
v = 0,u=W + a,b = a = h/2, A =$ (§ , Pt = $ (qi'} , and
\ W \(±i/
we obtain
; 2r'
i _ i _ Vf .(S)2 / W . T' I yj PA
2£©2 (0 ; T | APt) 7?t ( ' ' l)'
i
where (7 is independent of p. It is assumed that the sum 2£®2(0; T' 4P;)
i
is different from zero for each of the 2? sets of values of the fourth roots &.
This formula suffices to express any theta function of the second order with
zero characteristic.
When (K, K'} is other than zero, by putting in the equation (I), § 367,
r = 2, /i=0, adding %rh' to F, where h' is a row of quantities each either
0 or 1, and then changing the sign of W, we obtain
Ce-**(K+M V2(W; K,K' + h') = 2 [e9^'^®' (W + a) + ee-^'^& ( W - a)],
h
where X = K + h, X' = K' + h', and C is the same constant as before, indepen
dent of W, K, K', h', and a = ±\ + £T'\', the period T' being omitted on the
right side. Hence, taking the fundamental formula of § 309 (p. 510), putting
therein t>=0, u=W+a, b = a, A=(),B=A, and then writing a-JX+|TV+4«,
where # is a row of p equal quantities, we find, provided | K, Pf | = 0, (mod. 2),
When (K, K') is zero, the function is necessarily even (§ 288, p. 463), and therefore |-fiT|=|Q|
We have seen (§ 327, Chap. XVIII.) that this is always true when r is odd. When r is 2, it is not
always so, as is obvious by considering the transformation, for p = l, in which a = 2, /3 = 0, a' = 0
/3' = 1, and ((?,<?') = (i,i); then we find (A', K') = (J, 1) ; thus |Q| = 1, \K\ = 2.
G18 GENERAL FORM FOR TRANSFORMATION OF SECOND ORDER. [370
and €==en\K\ + ni\A,K\t that 2*W2 (W ; K, K' + li) is equal to the limit,
when a; vanishes, of the expression
a;\A,K, Pi)
+ ®(W-x\A,K,Pi)},
U y (P*\ r
where c; i = { A } e
\ ^i_ /
i (JlQ -~~ Itf Q-i c . o Y) t i
• 1 Jl' Cj. ( 1 1 M 1
x-*"1- /
' ! i /''A P ^
2gi«
1 — 2
(0; A, Pi)
i
It can easily be proved (cf. § 308, p. 510) that the denominator of Ef
vanishes, for x = 0, for the 2^ sets of values of the fourth roots & in which
the fourth root corresponding to the characteristic K of the group (P) has
(A\
) ei*l*l, and that the corresponding expressions
L, K, Pi) + S(W-x\A, K,Pt)}
have the limit zero ; the summation 2 is therefore to be taken only to extend
i
A.
(.\
Kje^ilKl. It may
however happen that the denominator of Ef vanishes for other sets of values
of the fourth roots &, when a?= 0. We assume that for such sets of values
the sum multiplying Ef in the expression Uf does not vanish for x = 0 ; by
recurring to the proof of the formula of § 308, it is immediately seen that
this is equivalent to assuming that the expression
; Pi)
is not zero for general values of the arguments U for any set of values of the
fourth roots e* (cf. (/S), p. 514). That being so, the value of Ef when its
denominator vanishes for a? = 0, can always be obtained from the limiting
expression given, by expanding its numerator and denominator in powers
of x.
Ex. Applying the formula of this page for the case p = 1 to the function
eu(2TF; 2rO = i*a(TP; -£, 1),
for which (A', £"') = (-£, 0) and A' = l, we immediately find that the Gopel system iu terms
of which the function can be expressed is (A, APJ, where A =£ ( Qj , Pl = K=$ ( _v) « we
/A\
are to exclude the value of the expression Uf in which { 1 = - f_J = 1 ; the value of Ef for
^= — 1 is easily found to be
872] THE PROBLEM OF THE CONSTANT COEFFICIENTS. 619
of which both numerator and denominator vanish for # = 0. The final result of the
formula is
Cen(2Tf; 2r')=-4eio(i; ^e^i; r')eu(JT; r'}Qw(W- r')/e'n (0; r')ej(J(0 ; r').
Prove this result, and also
<7eol(2Tf; 2r') = 2e^(i; r>)eM(W; r')eoi(Tf; 0/6^(0; r') 901 (0 ; r'),
and (cf. § 365) obtain the formulae
or
A
or w
K
e,(i; r'Hie^O; r')e01(o; r')[e^(0; r')+e^(o; r')],
ej,(0j 20 = 4 [ej,(0; 0 + 6^(0:0],
(7= V2[eJ,(0; 0 + 8^(0; r')].
371. The preceding investigations of this chapter enable us to specify in
all cases the form of the function §(u: 2o>, 2&>' 2??, 2??' ^ )
V QJ
when expressed in terms of functions ^ (w ; 2i/, 2i/. 2(T,
\
In many particular cases it is convenient to start from this form and
determine the coefficients in the expression by particular methods. But it
is proper to give a general method. For this purpose we should consider
two stages, (i) the determination of the coefficients in the expression of the
/ | Q'\
function ^ (u L J by means of functions tyr (w ; K, K' + /&), (ii) the determi-
\ c /
nation of the coefficients in the expression of the functions i/rr (w ; K, K' + //,)
(«*'\
w \. The preceding formulae of this chapter
enable us to give a complete determination of the latter coefficients in a
particular form, namely, in terms of theta functions whose arguments are
fractional parts of the periods 2v, 2v ; but this is by no means to be regarded
as the final form.
372. Dealing first with the coefficients in the expression of the function
*\ Q) b^ functions $r(w, K,K' + (JL), there is one case in which no
difficulty arises, namely, when the transformation is that associated with the
matrix ^ J ; then S- (u I Q ) is equal to S (rw ; 2v, 2rv', 2f/r, 2^' K'^'\ ,
the row K' being in fact equal to rQ', namely * (u Q} is ^,. (w ; K, K').
620 THE FIRST STEP IN THE [372
Now it can be shewn*, that if ftr be the matrix associated with any
transformation of order r, and r be a prime number, or a number without
square factors, then linear transformations, ft, ft', can be determined such
(T 0\
that ft,. = ft f j ft . Hence, in cases in which the matrices ft, ft' have been
calculated, it is sufficient, first to carry out the transformation ft upon the
given function Sfcfti j; then to use the formulae for the transformation
(r 0\
j , whereby the original function appears as an integral polynomial of
order r in 2^ theta functions ; and finally to apply the transformation ft' to
these 2f theta functions. All cases in which the order of transformation is
not a prime number may be reduced to successive transformations of prime
order (§ 332, Chap. XVITL).
We can however make a statement of greater practical use, as follows. It
is shewn in the Appendix II. (§§ 415, 416) that the matrix associated with
any transformation of order r can be put into the form ft ( „, j , where ft
is the matrix of a linear transformation, and that, in whichever of the possible
ways this is done, the determinant of the matrix B' is the same for all. In
all cases in which this has been done the required coefficients are given by
the equation
• ,,
V|ft)| |
W+^'.Se-->->. & r . 2v_ w> 2?/). 2?, <A"+M)/r1 1
H K
^ \M\\v\\R
wherein, (Q, Q') being a half-integer characteristic, e is an eighth root of unity,
u = Mw, \M\ is the determinant of the matrix M, etc., /m is in turn every
row of integers each positive (or zero) and less than r, which satisfies the
condition that the p quantities - B'/j, are integral, and, finally, 7 denotes the
symmetrical matrix BB', while d denotes the row of integers formed by the
diagonal elements of 7. It is shewn in the Appendix II., that the resulting
range of values for p is independent of how the original matrix is resolved
into the form in question. For any specified form of the linear transformation
ft the value of e can be calculated (as in Chap. XVIII., §§ 333—4); if e0
* Cf. Appendix II.; and for details in regard to the case^ = 3, Weber, Ann. d. Mat., Ser. 2a,
t. ix. (1878—9). We have shewn (Chap. XVIII., § 324, Ex. i.) that the determinant of the
matrix of transformation is ±?-P. From the result quoted here it follows that that determinant
is +7*.
372] DETERMINATION OF THE COEFFICIENTS. 621
denote its value when the characteristic (Q, Q') is zero, its value for any other
characteristic is given by
where H = , and Q,' = pQ' - p'Q-^d(pp'), - Q, =5= Q' - <f'Q -%d(<r<r').
To prove this formula, we have first (§ 335, Chap. XVIII.), if fl =
P
the equation
= *(u; 2(o, 2ft)',
Q'
\f\J\f,
where u = M1ul, Mlwl = cap + w'p, etc. Writing ul = 2o)1U'l, WI/ = O)ITI, we
have
j (u^ 5 2o)i 2(&i , 2f]i, 2i)i
and the equations w.j = J/2w, Jl/2i; = w^A, M^v = ^B + w^B', give, if w = 2vW,
v = vr\ and in virtue of AB' — r, the equations Ul = A W, rr^ = AT' A — BA,
while, by the equation r£= M^A, we find r}l(o1~1u12 = r%u~lw*. Now it is
(o
Ul ; X
gives
( m + - }
\ r/
ym» + dm) -
r
wherein 7 = 55', and cZ denotes the row of diagonal elements of 7, and m, p,
are obtained by putting An—- rm + p, m being a row of integers, and /* a row
of integers each less than r and positive (including zero) ; this equation is
equivalent to n — B'm = -B'ij,; corresponding to every n it determines an
unique m and an unique p for which — - is integral ; corresponding to any
TV
assigned p, for which - is integral, and an assigned m, the equation
determines an unique n. Since then yw2 + dm is an even integer, and, for
the terms which occur, B — m is an integer, we have
Increasing, in this equation, U, by Ql + TjQ/, we hence deduce
; T,
e
r
[r!T;
622 A PARTICULAR EXAMPLE OF THE SECOND STEP [372
where K' = AQ,', - K = BQ1' - B'Q1-^d(BB')i so that (K, K') is the
characteristic of the final theta function of w. Since now the matrix
MvB' = M.M^B' = M^AB' = rMlwl , and therefore \M\ v\ B'\ = rP\Ml mi\,
we have, by multiplying the last obtained equation by e^i<ai u>* = e%r&~lw2 , the
formula which was given above.
Ex. i. When p = l, the transformation associated with the matrix ( ) gives rise to
\ J
the function 9(TF ; \T} ; we have
©/ TI7 . 1 _'\ o /Q H7 • Q— '\ l_ £i I O T'lT . O '
^ rr j >rT ^ — O ^O rr ^ Or y ~J~ v7 I «5 rr j OT
V
Other simple examples have already occurred for the quadric transformations (§ 365).
Ex. ii. Prove when p = 2, by considering the transformation of order r (r odd) for
which
,0, */• vo, oy vo o.
that
6 it,— u.u<>. ru9\ - (T,, — 2ur19 + u2r.,.> — 2X), 2r,9 — 2uT9,, rrool
' ^«' j^^,\H I li I HI /7 \.£t |~ ^^ 7 i«J
', 0)+S 2
where ^(OTI, 74) denotes B^J rr
+ e
y . (Wiltheiss,
Crelle, xcvi. (1884), pp. 21, 22.)
373. In regard now to the question of the coefficients which enter in the
/ Kf"
expression of the functions i/>y (w ; K, K' + /i) by means of functions S- ( w r ) ,
\ K )
the problem that arises is that of the determination of these coefficients in
terms of given constants, as for instance the zero values of the original theta
functions. The theory of this determination must be omitted from the
present volume. In the case when the order of the transformation is odd
these coefficients arise in this chapter expressed in terms of theta functions,
»M~ 5 2t>, 2i/, 2£, 2^'J , whose arguments are ?'-th parts of the
periods 2i», 2i/. By means of two supplementary transformations, A, 7'A"1,
(as indicated § 332, Chap. XVIII.), or by means of the formulae of Chap. XVII.
(as indicated in Ex. vii., § 317, Chap. XVII.), we can obtain equations for
functions ^ (rw ; 2u, 2i/, 2f, 2£") as integral polynomials of degree r2 in
functions S-(w; 2v, 2t/, 2£ 2£"). By means of these equations the functions
^•(— -; 2u, 2i/, 2£, 2£'j are determined in terms of functions
^ (0 ; 2u, 2i/, 2f, 2^') ; or this determination may arise by elimination from
the original equations of transformation, without use of the multiplication
equations. There remains then further the theory of the relations connecting
the functions &(0; 2u, 2i/, 2£ 2f) and the functions ^(0; 2eo, 2ft)', 277, 2?/),
which is itself a matter of complexity.
373] IN THE EXPRESSION OF THE CONSTANT COEFFICIENTS. 623
For the case/>=l, the reader may consult, for instance, Weber, Elliptische Functionen
(Braunschweig, 1891), Krause, Theorie der doppeltperiodischen Functionen (Erster Band,
Leipzig, 1895). For the case jo = 2, Krause, Hyperelliptische Functionen (Leipzig, 1886),
Konigsberger, Crelle, LXIV., LXV., LXVII. For the form of the general results, the chapter,
Die Theilung, of Clebsch u. Gordan, AbeUsche Functionen (Leipzig, 1866), which deals with
the theta functions arising on a Riemaun surface, may be consulted. For the hyper-
elliptic case, see also Jordan, Traite" des Substitutions (Paris, 1870), p. 365, and Burkhardt,
Math. Annal. xxxv., xxxvi., xxxvm. (1890 — 1).
In particular cases, knowing the form of the expression of the functions
3 (u ; 2o>, 2«', 2r], 2rj')
in terms of functions 3 (w ; 2v, 2i/, 2£, 2f), we are able to determine the coefficients by the
substitution of half-periods coupled with expansion of the functions in powers of the
arguments. See, for instance, the book of Krause (Hyperelliptische Functionen} and
Konigsberger, as above.
Ex. i. In case p = 2, r = 3, the function 05(3JF, 3r') is a cubic polynomial of the
functions 06 ( W, T'), 634 ( W, T'), 0t ( Wt T'), e02 ( W, r'), of which the characteristics are
respectively £L' QJ, $(' _ V J/_ _\ $( ' QJ ; these form a Gopel system.
The only products of these functions which are theta functions of the third order and of
zero characteristic are those contained in the equation
©5 (3 W, 3r') = Atl + B<t>5$l4 + CWl+D<t>^l2 + E<p3t4>1^
where $6 = 05(JF, T'), etc.; this equation contains the right number £ (rp + 1 ) = 5 of terms
on the right side. Putting instead of the arguments W1, TF2 respectively
we obtain in turn
eM(3lF, 3r')= A^
0, (Sir, 3/)= -A<fi
eM(3Tr, 30= -^L
whereby the Gopel system of functions 06 (3 W, 3r), ©34 (3 W, 3r'), 6X (3 W, 3r'), e02 (3 IF, 3r')
is expressed by means of the Gopel system <£5, ^34, <£j, ^02.
From the first two equations, by putting the arguments zero, we obtain
®/;efi-Q<ue<u
t _ 65 3434 n _
•** ~~ 4 4 9 —
e5~634 95e34 (66 ~ 934
where 05 = 05(0 ; 3r'), etc., and 0. = 05(0 ; T'), etc. ; by the addition of other even half-
periods to the arguments, for instance, those associated with the characteristics
O,O' -i,
we can obtain expressions for C, D, E ; these substitutions give respectively
0,3 (3 IK ; 3r') = ^<#
04 (311'; 3r') = A<f
0,, (3 IF ; 3r') = .1 0J.,
624 PARTICULAR EXAMPLE. [373
putting herein W=0 we obtain in succession the values of Z>, C and E, expressed in terms
of the constants previously used, 05, 034, 65, 634 and the constants 023, ©4, 012, G^, 003,
04J eM, 012, 00, 62, 001. Thus the zero values of each of the ten even functions 0( W \ r)
enter in the expression of the coefficients J, B, C, D, E ; there remains then the question
of the expression of the zero values of the ten even functions in terms of four independent
quantities (cf. Ex. iv., § 317, Chap. XVII.), and the question of the relations connecting
the constants 05, ©34, etc., and the constants 05, 034, etc. (cf. the following example).
Ex. ii. Denoting 001 (0 ; 3r') 001 (0 ; r') by <701, etc., shew that when p = 2 the result of
Ex. iii., § 292 (p. 477) gives the equations
^01 + ^2 = £5 + ^34 ~ ^12 ~ ^0 J
^4 + ^03 = ^5 ~ ^34 + ^12 ~~ ^0>
^23 + ^14 = ^5 ~ ^34 ~ ^12 + ^OJ
these being the only equations derivable from that result. By these equations, in virtue of
the relations connecting the ten constants 0 (0 ; r'), and the relations connecting the ten
constants 0 (0 ; 3r'), (for the various even characteristics), the three ratios
634(0; 3r')/05 (0, 3r'), 012(0; 3r')/e5(0; 3r'), 00 (0 ; 3r')/66 (0 ; 3r')
are determinate in terms of the three
034 (0 ; r')/e5 (0 ; T'), 012 (0, r')/05 (0 ; r'), 00 (0 ; r')/66 (0 ; r').
By addition of these equations we obtain
^01 + ^2 + ^4 + ^03 + ^23 + ^14 + ^34 + ^12 + ^0 = ^ Cfi •
Obtain similarly from the result of Ex. iii., § 292, for any value of p, the equation
20^0; 3r' i(j')]e[°; ' i(j')] = (2p-l)e(0; 3r') 6 (0 ; r'),
where the summation on the left extends to all even characteristics except the zero
characteristic ; for instance, when p — I, this is the equation
001 (0 ; 3r') 001 (0 ; r'} + 610 (0 ; 3r') 610 (0 ; T') = Qw (0 ; 3r') 0^ (0 ; r'),
namely (cf. Ex. i., § 365 of this chapter) it is the modular equation for transformation of
the third order which is generally written in the form (Cayley, Elliptic Functions, 1876,
p. 188),
As here in the case j» = 2, so for any value of p, we obtain, from the result of Ex. iii.,
§ 292, 2P— 1 modular equations for the cubic transformation.
Ex. iii. From the formula of § 364 we obtain modular equations for the quadric
transformation, in the form
where s is a row of p quantities each either 0 or 1, so that the right side contains 2" terms,
and k, k', s' are any rows of p quantities each either 0 or 1.
374. In the fundamental equations of transformation we have considered
only the case when the matrices a, a', /3, /3' are matrices of integers ; the
analytical theory can be formulated in a more general way, as follows; the
argument is an application of the results of Chap. XIX.
374] GENERALISED FORM OF THE EQUATIONS OF TRANSFORMATION. 625
Suppose we have the relations expressed (of. Ex. ii., § 324, Chap. XVIII.)
by
( M, 0 ) ( 2v, 2i/ ) = ( 2&), 2o>' ) ( a , /3 ),
0 , rJf-1 I 2£ 2£' 277, 277' , a.', 0'
where r is a positive rational number, M is any matrix of p rows and columns,
whose determinant does not vanish, a, /3, a', /3' are matrices of p rows and
columns whose elements are rational numbers not necessarily integers, &>, w ' ,
V), ?;' are matrices of p rows and columns satisfying the equations (B), § 140
(Chap. VII.), and v, v', f, £" are similar matrices satisfying similar conditions ;
then, as necessarily follows, the matrices a, @, a', $' satisfy the relation
(viii) of § 324 (Chap. XVIII.).
If now x, y be any matrices of p rows and columns, the relations supposed
are immediately seen to be equivalent to
(M, 0
0 , rM~
we suppose that x, y are such matrices of integers that ax, fty, ax, ft'y are
matrices of integers, and, at the same time, such that rx is a matrix of integers ;
such matrices x, y can be determined in an infinite number of ways.
Let u, w be two rows of p arguments connected by the equations u = Mw ;
when the arguments w are simultaneously increased by the elements of the
row of quantities denoted by 2vxm + 2v'ymf, in which m, m' are rows of p
integers, the arguments u are increased by the elements of the row 2&>r? + 2<»W,
where n = axm + &ym, n = a'xm + ft'ym' are rows of integers. The resulting
factor of the function *&(u; 2&>, 2o>', 2?7, 277') is eR, where, if Ha = 2r)a+ ZrfoL,
etc., (cf. (v), § 324, Chap. XVIII.), R is given by
R = Hn (u + ^ Hn) — irinri
= (Haxm + Hpym') (Mw + Mvxm + Mv'ym'} — -rrinn
= (MHaxm + MHpym'} (w + vxm + vym) — Trinri
= r (%%xm + 2^'ym') (w + vxm + vym'} — trinn ;
now, since J3'a = r + /3a, and because ax, fty, OL'X, ft'y, rx are matrices of
integers, we have
nn' = xa'axm? + (yJ3a'x + y^ax) mm + yft'Pym"1
=fm +f'm' + ryxmm (mod. 2),
where /, /' denote respectively the rows of integers formed by the diagonal
elements of the symmetrical matrices XOL'OLX, yP'fty (cf. § 327, Chap. XVIII.).
Thus, if we denote ^ (u ; 2<w, 2<o', 2?7, 277') by <f> (w), we have
(w + 2vxm + 2v'ym') = er{'^xm+*l>'ym'> (w+vam+w'j/w') +«' (/»»+/•»»') +Tn(
B- 40
628
THE TRANSFORMATION OF THETA FUNCTIONS
[374
Further if a, b denote the matrices of Zp columns and p rows, given
respectively by
a = (2ws, 2v'y), Zirib = (2r(fo 2r£'y),
we have
fjr^ —
% — (ab- "0 =
y ?
= ( a; (v £-£«)«,
0, -x
yoc, 0
so that a& — 6a = k, say, is a skew symmetrical matrix of integers given by
ab — ba = k = ( 0 , —rxy ),
ryx, 0
and we have
a</3
2 &„ m
= — ryxmm',
Finally, let X, /x be rows of £> quantities, the rows of conjugate complex
quantities being denoted by X1; /ml} and let X, p be taken so that the row of
quantities a (X, p,) consists of zeros, or
a (X, /ji) = 2twX + %v'yp = 0,
so that af\ = — T'yfji, where* r =v~lv', is a symmetrical matrix, = p' + ia-', say,
p' and cr' being matrices of real quantities ; then by
we have
ik (X,
= - r
in which v = yp, ^ = y^ ; as in § 325, Chap. XVIII., since r is positive, the
form ra-'vv-i is necessarily positive except for zero values of p.
On the whole, comparing formula (II), § 354, Chap. XIX., the function
<f>(w) satisfies the conditions of §§ 351 — 2, Chap. XIX., necessary for a
Jacobian function of w in which the periods and characteristic are given -f- by
i«?, 2v'y),
e =
* The determinant of the matrix u is supposed other than zero, as in Chap. XVIII., § 324.
+ In § 351, Chap. XIX., the row letters have a elements ; in the present case a is equal to 2p,
and it is convenient to represent the corresponding row letters by two constituents, each of p
elements ; and similarly for the matrices of 2p columns and p rows.
374]
IS A CASE OF THE EXPRESSION OF JACOBIAN FUNCTIONS.
627
To this function we now apply the result of § 359, Chap. XIX., in order to
express it by theta functions of w. The condition for the matrix of integers
there denoted by g, namely geg = k, is satisfied by g = ( ' ) , for
v " > y i
( rx, 0 ) ( 0, -1 ) ( rx, 0 ) = ( rx, 0 ) ( 0, -y ) = ( 0 , -rxy)\
0, y 1,0
0, y
0, y rx, 0
ryx, 0
hence, with the notation of § 358, Chap. XIX.,
0 ]=(2v/r, 2t/),
o , r1
i
-X'
o U
0 , yr
Hence, as our final result, by § 359, Chap. XIX., the function </> (w), or
^ (u ; 2&), 2&)', 2?;, 277'), can be expressed as a sum of constant multiples of
functions* § (w ; 2v/r, 2i/, 2£ 2f ) m'«A different characteristics, the number of
such terms being at most VjlTi = T* # |y|t wAere [#(, |y| de,lote *Ae
determinants of the matrices x, y. This is an extension of the result
obtained when the matrices a, j3, a, j3' are formed with integers ; as in that
y\, owing
case there will be a reduction in the number of terms, from r*
to the fact that the function </> (w) is even. A similar result holds whatever
be the characteristic of the function ^ (u ; 2<w, 2<o', 277, 277'). The generalisa
tion is obtained quite differently by Prym and Krazer, Neue Grundlagen
einer Theorie der allgemeinen Thetafimctionen (Leipzig, 1892), Zweiter Theil,
which should be consulted.
Ex. Denoting by E the matrix of p rows and columns of which the elements are zero,
other than those in the diagonal, which are each unity, and taking for the matrices a, /3,
a', /^ respectively ^ E, 0, 0, — E, where m, n are integers without common factor, we have
the formula
n
« 2261—
*
ms/n\
nr/mj '
wherein r, s are rows of p positive integers, in which every element of r is 0 or numerically
less than m, and every element of s is 0 or numerically less than n. This formula includes
that of § 284, Ex. iii. (Chap. XV.) ; it is a particular case of a formula given by Prym and
Krazer (loc. cit., p. 77).
To obtain a verification— the general term of the right side is e*, where
40—2
That is, functions 3 (rw, 2v, 2ri/, 2f/r, 2f ) ; cf. § 284, p. 448.
628 THE ALGEBRAICAL APPLICATIONS OF THE THEORY. [374
hence 26* = 0 unless N/m is integral ; when N/m is integral, =M, say, then 2e* = ?»pe*,
r r
where
K, =nM+s, obtaining all integral values when M takes all integral values and s takes all
integral values (including zero) which are numerically less than n.
375. The theory of the transformation of theta functions may be said to
have arisen in the problem of the algebraical transformation of the hyper-
elliptic theta quotients considered in Chap. XL of this volume. To practically
utilise the results of this chapter for that problem it is necessary to adopt
conventions sufficient to determine the constant factors occurring in the
algebraic expression of these theta quotients (cf. §§ 212, 213), and to define
the arguments of the theta functions in an algebraical way. The reader is
referred* to the forthcoming volumes of Weierstrass's lectures.
It has already (§ 174, p. 248) been remarked that when p>3 the most
general theta function cannot be regarded as arising from a Riemann
surface ; for the algebraical problems then arising the reader is referred
to the recent papers of Schottky and Frobenius (Crelle, Gil. (1888), and
following) and to the book of Wirtinger, Untersuchungen uber Thetafunctionen
(Leipzig, 1895).
* Cf. Rosenhain, Mem. p. divers Savants, xi. (1851), p. 416 ft.; Konigsberger, Crelle, LXIV.
(1865), etc.
377]
CHAPTER XXI.
COMPLEX MULTIPLICATION OF THETA FUNCTIONS. CORRESPONDENCE OF
POINTS ON A RIEMANN SURFACE.
376. IN the present chapter some account is given of two theories ; the
former is a particular case of the theory of transformation of theta functions ;
the latter is intimately related with the theory of transformation of Riemann
theta functions. Not much more of the results of these theories is given
than is necessary to classify the references which are given to the literature.
377. In the transformation of the function © (u; T), to a function of the
arguments w, with period r' (§ 324, Chap. XVIII.), the following equations
have arisen
u = Mw, M=OL + TO.', Mr' = j3 + r/3';
there* are cases, for special values of r, in which T' is equal to r. We
investigate necessary conditions for this to be so ; and we prove, under a
certain hypothesis, that they are sufficient. The results are stated in terms
of the matrix of integers associated with the transformation ; we do not enter
into the investigation of the values of r to which the results lead. We limit
ourselves throughout to the function (B) (u ; r) ; the change to the function
^ (u ; 2o>, 2o)', 2?7, 2?/) can easily be made.
Suppose that, corresponding to a matrix A = ( , ^ J , of 2p rows and
\W, fij /
columns, for which
a/8 = £a, a'/8' = /S'a', a£' - £*' = r = ffa. - a'/S,
where r is a positive integer, there exists a matrix T satisfying the equation
(a + ra') r = £ + r/3',
which is such that, for real values of nl} ..., np, the imaginary part of the
quadratic form rn2 is positive.
* References to the literature for the case^ = l are given below (§ 383). For higher values of
p, see Kronecker, Berlin. Monatsber. 1866, p. 597, or Werke, Bd. i. (Leipzig, 1895), p. 146;
Weber, Ann. d. Mat., Ser. 2, t. ix. (1878—9), p. 140; Frobenius, Crelle, xcv. (1883), p. 281,
where other references are given ; Wiltheiss, Bestimmung Abehcher Funktionen mit zwei
Argumenten u. B. w. Habilitationsschrift, Halle, 1881 (E. Karras), and Math. Annal. xxvi.
(1886), p. 130.
630 STATEMENT AND PROOF OF THE NECESSARY [377
In that case, as follows from Chap. XX., the function ® [(a + rot') w ; T],
when multiplied by a certain exponential of the form e?™*, is expressible as an
integral polynomial of the r-th degree in 2p functions ® [w ; T] ; on this
account we say that there exists a complex multiplication*, or a special
transformation, belonging to the matrix A. The equation (a + TO?)T = /3+T/3'
is equivalent to (/•?' — TO!) r = — ft + TO. ; this arises from the supplementary
matrix
just as the former equation arises from A ; we put M = a + TOE', N = ft' — ra ;
we denote by A — X the determinant of the matrix A — \E, where E is the
matrix unity of 2p rows and columns, and X is a single quantity ; similarly we
denote by M — X the determinant of the matrix M — XJE", where E' is the
matrix unity of p rows and columns.
Then we prove first, that when there exists such a complex multiplication,
to every root of the equation in X of order p given by \ M — X =0, there
corresponds a conjugate complex root of the equation N — X | = 0 ; that the Zp
roots of the equation A — X =0 are constituted by the roots of the two equations
|lf-X| = 0, N-\\ = 0, or A-X = M — \ \N-\\; and that all these
roots are of modulus *Jr. Hence when r = 1, they can be shewn to be all
roots of unity.
378. The equations of the general transformation, of order r, and its supplementary
transformation, namely
M
give
hence, if r=r1+^r2, where rl and r.2 are matrices of real quantities, and similarly r — i-/ + tV2',
we have by equating imaginary parts
(a + T^G') T2' = r2 (& ~ a'ri') J
therefore the two matrices
) r/ + tV2 aV2', r2 ^V= r2 (ft — a'r/) — zV2 aV2'
are conjugate irnaginaries, =f+ig and/— igt say.
Now suppose T' = T ; then from
MTZ =/+ ifft T2F=/- iff,
we have, if X be any single quantity, and J/0 be the matrix whose elements are the
conjugate complexes of the elements of J/,
(J/0 - X) r2 =/- ig - Xr2 = r2 (N - X),
and hence, as | r2 1 is not zero,
|jr.-xf-|jr-u
* The name principale Transformation has been used (Frobenius, Crelle, xcv.).
378] CONDITIONS FOR THE MATRIX OF TRANSFORMATION. 631
which shews that to any root of the equation \M-\ \ = 0 there corresponds a conjugate
complex root of the equation | /V-X | =0. Further we have, if r0 = r1 — iY2,
/I r \ /a 0\ = /J/ #r \ /J/ 0 \ /I
VI rj U 07 " Wo JftfiJ V 0 #o/ V
and writing this equation in the form
where
1
1
it easily follows that the determinant of the matrix t is not zero, and that, if X be any
single quantity, we have
so that
I A \ , I \ I Tf \ I I/" \ I I \f \ I I AT" "\
A — A ^ Lt — A == JJL — A Jfl Q — A — ( JJ1 — A | XT — A
Thus the roots of the equation | A - X ' = 0 are constituted by the roots of the equations
Further, from a result previously obtained (Chap. XVIII., § 325, Ex.), when, as
here, T' = T and 2<u = l, 2v = l, we have
also as, for real values of %, ..., np, the form r2?i2 is a positive form, it can be put into the
shape mj* + ...... +*£, =Emz, say, ^ being the matrix unity of p rows and columns, and
m being a row of quantities given by m = Sn, where S is a matrix of real elements ; the
equation rinz = E. Sn. Sn gives Tz = tiES=X8 ; for distinctness we shall write
r2 = AYf0,
K=K0 = S being conjugate complex matrices. Take now a matrix R = KMK~l ; then
K'1 - K~ lMrMK~l = rK~ K~l = r ;
thus if X be a root of j M—\ =0, and therefore, as R-\ = K (M-\~) K~l, also a root of
R-\ | =0, and if 2, =z + iy, be a row of p quantities such that Rz=\z=E\z, where E is
the matrix unity of p rows and columns, we have
SiRfyz = R0z0. Rz — E\0z0 . E\z = XX0 . Ez0z
or
Therefore as Ez9z, which is equal to 2 (^m+y2m), is not zero, it follows that XX0 = r; in
m=l
other words, all the roots of the equations ; M— X =0, | A - X | = 0, are of modulus Jr.
Suppose now that r = l, so that the roots of the equation | A — X | = 0 are all of modulus
unity ; then we prove for an equation
of any order, wherein the coefficients .1, /?, ..., AV are rational integers, and the coefficient
of the highest power of x is unity, that if all the roots be of modulus unity, they are also
roots of unity* ; so that, for instance, there is no root of the form elV2.
* Kronecker, Crelle, LIII. (1857), p. 173; Werke, Bd. i. (1895), p. 103.
632 NECESSARY CONDITIONS. [378
Let the roots be e1", e^, ..., so that
A= - (cos a+cos/3 + ...), # = cos
then A lies between - n and n, and B lies between ±\n (n - 1), etc. ; hence there can only
be a finite number, say k, of equations of the above form, whereof all the roots are roots of
unity. Thus, if xlt ..., xn be the roots of our equation, so that, for any positive integer /*,
the roots of the equation
are also roots of unity, it follows that, of the equations
^0*0 = 0, ^2(a;) = 0, ..., Ft + l(x) = 0,
there must be two at least which are identical. Hence, supposing F (x) = 0, F (x) = 0 to
be identical, we have n equations of the form
fX. _ V fl _ V
x\~xrj •****%' ••"
Choosing from these equations the cycle given by
i~ r,' r, «,'""' nti~ l'
consisting, suppose, of o- equations, we infer that
and, hence, that x± is a (p* — v°")-th root of unity.
Ex. Prove that, when M=a + ra, .¥r' = /3 + 7-/3',
/ir\/« 0\ /jrowiyx
Vl Tj\J ft) VO JfoAl rj'
and deduce*, if A = f a, ' ) and
\* P /
'1 0
that
Hence, when T' = T, if s be a row of 2jt? elements, and # = A?, we have
which expresses a self- transformation of the quadratic form Hz2, which has real coefficients.
Cf. Hermite, Compt. Rendus, XL. (1855), p. 785 ; Laguerre, Journ. de I'ec. pol., t. xxv.,
cah. XLII. (1867), p. 215 ; Frobenius, Crelle, xcv. (1883), p. 285.
379. Conversely, let
be a matrix of integers of 2p rows and columns, such that
aa' = a'a, J3/3' = J3'j3, a/3' - a/3 = r = yS'a - /ffa',
* Cf. Chap. XVHI. § 325, Ex.
381] SUFFICIENT CONDITIONS FOR A COMPLEX MULTIPLICATION. 633
where r is a positive integer ; and suppose that the roots of the equation
| A — \ = 0 are all complex and of modulus \Jr. Under the special
hypothesis* that the roots of \ A — A, =0 are all different, we prove now that
a matrix r can be determined such that (i) r is a symmetrical matrix, (ii) for
real values of nlt ..., np the imaginary part of the quadratic form rri* is
positive, (iii) the equation
(a + rat) r = /3 + r/3'
is satisfied. Thus every such matrix A gives rise to a complex multiplication.
380. We utilise the following lemma, of which we give the proof at once. — If A be a
matrix of n rows and columns, such that the determinant |A + X , wherein X is a single
quantity, vanishes to the first order when X vanishes, and if #, y be rows of n quantities
other than zero, such that
hx=0, hy=Q,
then the quantity xy, =x1yl + ...... + xnyn, is not zero.
Denoting the row x by £t, its elements being £n, ..., £ln, determine other n(n-l)
quantities £i>;- (i=2, ..., n ; _/=!,..., n) such that the determinant |£ | does not vanish ;
similarly, denoting y by TJI} determine n(n-\) further quantities i^y such that the
determinant | rj \ does not vanish. Then consider the determinant of the matrix rj (h + A) £ ;
the (r, s)-th element of this matrix is
2 rjr> i 2 Ai, >£«,.,• + X 2 77,., i£gj i = 2 £,, y 2 /^ y^,-, f + X 2 rjri ^r> iy
i j i .1 i i
(i=l, ...,»; _/=!, ..., «), and when r=l we have
2Ai
1
while when 8 = 1, we have
2^,>&,,-=Wi,1 + ...... +/*i,»«i,n=(^)«=0;
thus the (1, l)-th element of this matrix is \xy, and every other element in the first row
and column has the factor X ; thus the determinant of the matrix is of the form X [Axy + \B\.
But the determinant of the matrix is equal to|A+X||£||?7|, and therefore by hypothesis
vanishes only to the first order when X vanishes. Thus xy is not zero.
381. Suppose now that X, X0, /*, ^0, ... are the roots of the equation | A — A | =0, where
X and X0, and p and /*„, etc. are conjugate complexes. It is possible to find two rows x, a/,
each of p quantities, to satisfy the equations
x' = \x', or, say, (A-A)(#, tf') = 0, (i),
and similarly two rows z, /, each of p quantities, to satisfy the equations
az + $z' = iLZ, a'z + P'z' = nz', (ii) ;
from equations (i), if o;0 be the conjugate imaginary to x, etc., it follows, since XX0=r, that
V T
' '
and hence, in virtue of the relations satisfied by the matrices a, /3, a', ft, we have
£'.r0 - j3.r0' = X.r0 , — a'^0 + ax0' = \x0' ,
* For the general case, see Frobenius, Crelle, xcv. (1883).
OF THF
UNIVERSITY
634 DETERMINATION OF A COMPLEX MULTIPLICATION [381
which belong to the supplementary matrix rA"1 just as the equations (i) belong to the
matrix A ; for our purpose however they are more conveniently stated by saying that
t = x0', t'= — x0, satisfy the equations
(A-A)(*, 0 = 0;
hence as x, x' satisfy the equations
(A-»(*,«0=0,
it follows from the lemma just proved, putting n = 2p, that tx + t'x' is not zero ; in other
words the quantity
is not zero. Further from the equations (i), (ii) we infer
\H (xz1 - x'z] = (ax + Qz') (a'z + /3Y) - (a'x + /3V) (as + /3/) ;
and by the equations satisfied by the matrices a, /3, a', ft this is easily found to be the
same as
(X/Li - r) (xzf - x'z] = 0 ;
thus, as the equation X/z = r would be the same as X = X0 , we have
xz' — x'z= 0.
Also we have
az0 + ftz0' = fi0z0, a'z0 + p'z0' = /i0 00' ;
thus we deduce, as in the case just taken, that
(Vo-^O^o'-AO^O;
and hence as X/*0 - r, =r (X/^i - 1), is not zero, we have
xz0' — x'z0 = 0.
If we put x=x1 + -ixz, x0=xl-ix2, x'=xl' + ix2', x§=x{-ix^, the quantity
xx0' — X'XQ = — 2i (x^ - x^x^)
is seen to be a pure imaginary ; if in equations (i) X be replaced by X0, the sign of xx^-x'x^
is changed, but the quantity is otherwise unaltered ; since then the equations (i) de
termine only the ratios of the constituents of the rows x, x', we may suppose the sign of
the imaginary part of X in equations (i), and the resulting values of the constituents of x and
x', to be so taken that
uCtJGfr 3G X{\ := — — ' *
this we shall suppose to be done ; and we shall suppose that the conditions for the (p— 1)
similar equations, such as
zz0'-z'z0= -2i,
are also satisfied. With this convention, let the constituents of x and x1 be denoted by
si, i > • • • > si. i» s i, 1 1 • • • > £ it P 5
similarly let the constituents of the rows 2, &', which are taken corresponding to the root p,
be denoted by
S2, 1> •••> C2,P> b2, 1> •••» ?2,P>
and so on for all the/> roots X, p, .... Then the equations xx0' - X'XQ = -2i, zz0' — z'z0= —2i,
etc., are all expressed by the statement that the diagonal elements of the matrix
are each equal to - 2i. When r is not equal to s (r, s<p + l), the (1, 2)-th element of this
matrix is
382] CORRESPONDING TO A GIVEN MATRIX OF PROPER FORM. 635
which we have shewn to be zero ; similarly every element of the matrix, other than a
diagonal element, is zero ; we may therefore write
Take now a row of p quantities, £, and define the rows X, X' by the equations
X=lt, X'=$t,
so that
^0 = bO^O) ^0=fo'o>
then
hence it follows that the determinant of the matrix % is not zero, since otherwise it would
be possible to determine t, with constituents other than zero, so that Jf' = 0, and therefore
also JT0'=0 ; as this would involve -2wy = 0, it is impossible.
382. If now the matrix T be determined from the equations
<u + TX' = 0, 2 + rz = 0, ... ,
where x, x are determined, as explained, from a proper value of A,, etc., or,
what is the same thing, if r be defined by
r+fr-o,
then
fF-rf-r^-r^-rV-W;
but the equations of the form xz — xz = Q are equivalent to
*F-ri=o;
now, since the determinant | £' does not vanish, a row of quantities t can be
determined so that X' = gt, for an arbitrary value of X' ; thus for this
arbitrary value we have
(T-r)Z/2 = 0,
and therefore
T = T,
or the matrix T is symmetrical.
Further, from the equation £ + £V = 0, we have
&' - rl. = rroio' - rr|o'= r (T, - T> g/,
and hence, if r = p + to-, since ££,' - f '|0 = - 2t, we have
I = f0io', or t0t = <rXQ'X',
where « is a row of any jt> quantities and X' = £'t ; hence, since the determi
nant g does not vanish, it follows, if X' be any row of p quantities, that
<rX0'X' is positive ; in particular when n1} ...,np are real, the imaginary part
of the quadratic form r?i2 is positive.
Finally from the equations
ax + fix' = \x, afx + fix = \x',
636 EXAMPLE OF COMPLEX MULTIPLICATION [382
putting x = — TX, we infer
(/8 — CUT) x — — \TX', (/3' — a'r) x = \x',
and therefore
T (/S' - err) X +(/3 — ar) a/ = 0,
or
[/8 + r/3' - (a + TO?) T] #' = 0,
and hence
[£ + r/3' - (« + ra') r] £' = 0,
from which, as | f ' j is not zero, we obtain
/3 + r/3' - (OL + ra.') T = 0.
We have therefore completely proved the theorem stated.
It may be noticed, as follows from the equation £ + fr=0, that we may form a theta
function with associated constants given by
2o> = 2£', 2co'=-2£;
these will then satisfy the equations
co'co — aw' = 0, o>o>o — o> O>Q — — 2* ;
the former equation always holds ; the matrix a> can be determined so that the latter
holds, as is easy to see.
Ex. Prove that by cogredient linear substitutions of the form
u' = cu, w' = cw,
we can reduce the equations u = Mw to the form
where p.lt ..., p.p are the roots of | M-\ =0.
383. For an example we may take the case p = l ; suppose that a, ft, a, ft' are such
integers that aft' - a'ft=r, a positive integer, and that the roots of the equation
are imaginary ; if a' = 0, the condition thatr should not be a rational fraction requires that
a /n_/«o\
a' ft')' W'
where a? = r, and then the equation for T is satisfied by all values of T ; this case is that of
a multiplication by the rational number a, and we may omit it here ; when a is not zero
we have _
2aV = - (a - ft') ± \/(a
and therefore (a+/3')2<4r; this of itself is sufficient to ensure that the roots of the
equation
are unequal, conjugate imaginaries, of modulus -Jr.
FOR THE ELLIPTIC FUNCTIONS.
637
383]
If then r be any given positive integer and h be a positive or negative integer
numerically less than 2jr, and a, a' be any integers such that (a2-£a + r)/a' is integral,
= - ft we obtain a special transformation corresponding to the matrix
for a value of r given by
a h —
A-2a
where | a' | is the absolute value of a', and the square root is to be taken positively ; the
corresponding value of M is a + ra'. Hence by the results of Chap. XX., the function
when multiplied by a certain exponential of the form exw*, is expressible as an integral
polynomial of order r in two functions 0 [w ; - ""^ I with different character
istics.
The expression for the elliptic functions is obtainable independently as in the general
case of transformation. When
Mv = <»a + a>'a, Mv = wj3 + <o'|3', ap>-a'@ = r, tl = Mw,
if to any two integers m, m! we make correspond two integers n, n' and two integers k, V,
each positive (or zero) and less than r, by means of the equations
rn + k = mfi - m'fi, rn' + k'=- ma + m'a,
or the equivalent equations
' = na' + rip' + - (a'
then we immediately infer from the equation
^(w) = tt-2+22'[(tt + 2m« + 2?ftV)-2-(2
m m'
by using n, ri, instead of m, m', as summation letters, that
2v,
wherein the summation refers to the r- 1 sets k, k' other than k = k'=Q, for which (§ 357,
p. 589) the congruences
ak + pk' = 0, a'k + pk' = 0 (mod. r)
are satisfied*.
This formula is immediately applicable to the case when there is a complex multiplica
tion ; we may then put
2o) = 2u=l, 2«' = 2w' = T, p = h-a, - ft = (a2 - ha + r)/a', r = (£-2a±zV4r-/i2)/2a',
* When these congruences have a solution (k0, kQ'), in which fr0, /r0' have no common factor,
i.e. (Appendix 11., § 418) when a, a', /3, ft' have no common factor, the remaining solutions are of
the form (XA-0, \k0'), where \<r; in that case taking integers x, x' such that k0x' - k0'x = l, it is
convenient to take 2vk0 + 2v'k0' and 2vx + 2v'x' as the periods of the functions g) on the right side.
638 EXAMPLE OF THE ELLIPTIC CASE. [383
and M—(h±i^4r-h2)/2, as above, where A2<4r. The application of the resulting
equation is sufficiently exemplified by the case of r — 2 given below (Exx. ii., in.).
In the particular case where r=l, the condition 7i2<4r shews that h can have only the
values 0 or + 1 or - 1 ; in this case the values «, n' given by
, ,,,,-, s
m — na + n (h — a)
a
n m and m' are integral;
immediately find
are integral when m and m' are integral; hence as — - -- " -- \-(k-a)T = MT, we
1
= 6022'
2 \°
= = ) 3V
*-*2'
7 -- u
n (ro+m'r)8
Thus when h = Q we have g3=0, and if a, a' be any integers such that (a2 + l)/a' is integral,
we have T=( + i—a)/a, the upper or lower sign being taken according as a is positive or
negative. In this case the function g> (u) satisfies the equation
(iW=4($to)
where
When h = l we have ff2 = 0, and if a, a' be any integers such that (a2-a + l)/o' is
integral, we have T = (l — 2a + ^\/3)/a' ; in this case
When h = - 1, we have <72 = 0, and, if (a2 + a + l)/a'be integral, then r- ( - 1 - 2a±t'V3)/a.
Ex. i. Denoting the general function <@u by ljf>(«; g%, <73), it is easy to prove that the
arc of the lemniscate r2 = a2 cos 2$ is given by a2//-2 = £> (s/a ; 4, 0) ; when n is any prime
number of the form 4& + 1 the problem of dividing the perimeter of the curve into n equal
parts is reducible to the solution of an equation of order k — when n is a prime number
of the form 2A + 1, the problem can be solved by the ruler and compass only. (Fagnano,
Produzioni Matematiche, (1716), Vol. u. ; Abel, CEuvres, 1881, t. I., p. 362, etc.) It is
also easy to prove that the arc of the curve i^ — a? cos 30 is given by a2/r2 = $(s/a; 0, 4);
when n is a prime number of the form 6^ + 1, the problem of dividing the perimeter of
this curve into n equal parts is reducible to the solution of an equation of order k (Kiepert,
Crelle, LXXIV. (1872), etc.). These facts are consequences of the linear special transforma
tions of the theta functions connected with the curves.
Ex. ii. In case r — 2, taking a = 4, a' = 9, A = 0, we have T = ( — l + i ^2)/9, and
By expanding this equation in powers of w, and equating the coefficients of w2, we
find easily that, if ft) (r/2) = e, then g2 = J^e2, and g3= — Je3; hence we infer that by means
of the transformation
o/:— r I ®
— ^$ — •tT"g / ,N
we obtain
dx
r ds i
h \/8P- 15£ + 7 J
which can be directly verified. It is manifest that when r = 2, h=Q, we are led to this
equation for all values of a and a.
CORRESPONDENCES ON A RIEMANN SURFACE. 639
Ex. iii. Prove that if m = \ (h + i \/8 - A2), the substitution
,3m*-3 1
m*£ = x-\ -- T — -. -- -=
m4 + 4 x-\
gives the equation
r *e _n r '. dx
Jl \/(m4 + 4)£3-15£-(m4-ll) ./* V(»4 +4) **-!»*-(«• -11)
This includes all such equations obtainable when r = 2. Complex multiplication arises
for the five cases /i = 0, h— + 1, h= ±2.
.Ek iv. When r-3 and_p=l, we see by considering the matrix
iwo -iwi i
that the function eia[(l+i \/2) w; i\/2] is expressible as a cubic polynomial in the
functions 80>1(w; t'\/2), elfi(w; i\/2). The actual form of this polynomial is calculable
by the formulae of Chap. XXI. (§§ 366, 372), by applying in order the linear substitutions
C 1%) ' (° " l} and then the cubic transformation (0 3) • Hence deduce that k = *J2-l
and
sn[(l + i\/2) TT] = (l+rV2)sn TT[l-sn2 F/sn2y]/[l-^2sn2 JF.sn2y],
where y = 2(K- i'A'')/3, K being (§ 365, Chap. XXI.) =7^, and iK' = rK.
For the complex multiplication of elliptic functions the following may be consulted :
Abel, (Euvres, t. I. (1881), p. 379 ; Jacobi, Werke, Bd. I., p. 491 ; Sohnke, Crelle, xvi.
(1837), p. 97 ; Jordan, Cours d' Analyse, t. II. (1894), p. 531 ; Weber, Elliptische Functionen
(1891), Dritter Theil ; Smith, Report on the Theory of Numbers, British Assoc. Reports,
1865, Part vi. ; Hermite, Theorie des equations modulaires (1859); Kronecker, Berlin.
Sitzungsber. (1857, 1862, 1863, 1883, etc.), Crelle, LVII. (1860) ; Joubert, Compt. Rendus,
t. L. (1860), p. 774; Pick, Math. Annal. xxv., xxvi. ; Kiepert, Math. Annal. xxvi. (1886),
xxxii. (1888), xxxvii., xxxix. ; Greenhill, Proc. Camb. Phil. Soc. iv., v. (1882—3), Quart.
Journal, xxil. (1887), Proc. Lond. Math. Soc. xix. (1888), xxi. (1890) ; Halphen, Liouville,
(1888); Weber, Ada Math. xi. (1887), Math. Annal. xxin., xxxin. (1889), XLIII. (1893);
Etc.
384. We come now to a different theory*, leading however in one phase
of it, to the fundamental equations which arise for the transformation of
theta functions, that namely of the correspondence of places on a Riemann
surface. The theory has a geometrical origin ; we shall therefore speak
either of a Riemann surface, or of the plane curve which may be supposed to
be represented by the equation associated with the Riemann surface, accord
ing to convenience. The nature of the points under consideration may
be illustrated by a simple example. If at a point a; of a curve the tangent
be drawn, intersecting the curve again in z1} z.,, ..., £n_2) we may say that to
the point x, regarded as a variable point, there correspond the n — 2 points
* For references to the literature of the geometrical theory, see below, § 387, Ex. iv., p. 647.
The theory is considered from the point of view of the theory of functions by Hurwitz, Math.
Annul, xxvin. (1887), p. 561; Math. Annal. xxxn. (1888), p. 290; Math. Annal. XLI. (1893),
p. 403. See also, Klein-Fricke, Modulfunctionen, Bd. n. (Leipzig, 1892), p. 518, and Klein, Ueber
Rienuinn't Theorie (Leipzig, 1882), p. 67. For (1, 1) correspondence in particular see the re
ferences given in § 393, p. 634.
640 STATEMENT OF NECESSARY CONDITIONS [384
z-i, . .., zn-2- To any point z of the curve, regarded as arising as one of a set
z-i, ..., zn-2, there will reciprocally correspond all the points, xl, x2, ..., x^^,
which are points of contact of tangents drawn to the curve from z. Such a
correspondence is described as an (n — 2, m — 2) correspondence. A point of
the curve for which x coincides with one of the points zlt ..., zn_2 correspond
ing to it, is called a coincidence ; such points are for instance the inflexions
of the curve.
In general an (r, s) correspondence on a Riemann surface involves that
any place x determines uniquely r places z1,...,zr, while any place z,
regarded as arising as one of a set zlt . .., zr, determines uniquely s places
x-i, ..., Xg. The investigation of the possible methods of this determination is
part of the problem.
385. Suppose such an (r, s) correspondence to exist ; let the positions of
z that correspond to any variable position of x be denoted by zlt ..., zr, and
the positions of x that correspond to any variable position of z be denoted by
x1, ..., xs; and denote by c1} ..., cr the positions of zlt ..., zr when x is at the
particular place a, and by a1} ..., as the positions of #1; ..., XB when z is at
the particular place c ; denoting linearly independent Riemann normal inte
grals of the first kind by vlt ..., vp, consider the sum
as a function of x ; since it is necessarily finite we clearly have equations of
the form
, r x, a -mf x, a Zj, cl zr, cr , • , %
Mi>lvl + ...... +Mi>pvp =vt + ...... +vt , (i = I,...,p\
where Miil} ...,Mijp are constants. On the dissected surface the omitted
aggregate of periods of the integral vt indicated by the sign = is self-deter
minative ; if the paths of integration be not restricted from crossing the
period loops the sign = can be replaced by the sign of equality (cf.
Chap. VIII. § 153, 158).
If now x describe the &th period loop of the second kind, from the right
to the left side of the kih period loop of the first kind, the places zlt ..., zr
will describe corresponding curves and eventually resume, in some order, the
places they originally occupied ; since, on the dissected Riemann surface
v*l>Cl -i- vf' Ca = VJ" Cl + v*i'C* , we may suppose each of them actually to resume
its original position ; hence we have an equation
wherein a^fc, «';,&, ... are integers ; similarly by taking x round the kth period
loop of the first kind we obtain
385] FOR AN (r, s) CORRESPONDENCE. 641
we have therefore 1p- equations expressible in the form
wherein a, a', /3, /3' are matrices of integers, of p rows and columns.
Consider next, as a function of x, the integral
/.'",('
vm
wherein z, c are, primarily, arbitrary positions, independent of x, and IIzj;CCi
is the Riemann normal integral of the third kind. The subject of integration
becomes infinite when any one of the places zl, ..., zr coincides with z, or, in
other words, when z is among the places corresponding to x, and this happens
when x is at any one of the places xlt ...,xg, which correspond to z\ the
subject of integration similarly becomes infinite when x is at any one of the
places a1} ..., ag, which correspond to the particular position of z denoted by c ;
it is assumed that c does not coincide with any one of the places cl5 ..., cr
The sum of the values obtained when the integral is taken, in regard to a,
round the infinities xly ..., xs, a1} ..., as, is, save for an additive aggregate*
of periods of the integral vm, equal to
This quantity is then equal to the value obtained when x is taken round
the period loops on the Riemann surface. Consider first, for the sake of
clearness, the contribution arising as sc describes the Mb. period loop of the
second kind ; if x described the left side of this period loop in the negative
direction, from the right to the left side of the Kh period loop of the first
kind, the aggregates of the paths described by z^...,zr would, in the
notation just previously adopted, be equivalent to aA)it negative circuits of
the \th period loop of the second kind, and a'Aj k positive circuits of the Xth
period loop of the first kind (X = 1, . . . , p). In the actual contour integration
under consideration the description by x of the left side of the Kb. period loop
of the second kind is to be in the positive direction ; hence the contribution
arising fur the integral as x describes both sides of the Kh period loop of the
second kind is
s.
- 2-TTlTm, k 2 a\, k V
?,c
A
similarly the contribution as x describes the sides of the kih period loop of
the first kind is
Which vanishes when paths can be drawn on the dissected surface connecting alt ..., a,
respectively to arlt ..., x,, so that simultaneous positions on these paths are simultaneous posi
tions of a-, , . . , x. . Cf . Chap. VIII. § 153 ; Chap. IX. § 165.
B- 41
642 ALGEBRAIC EXPRESSION OF CORRESPONDENCES [385
where Em> k = 0 unless m = k, and Em> m = 1. Taking therefore all the period
loops into consideration, that is, k = 1, ... , p, we obtain
«
where JVm, x = ft\, m - S Tm> k «'x, t •
*=i
so that #„,, x is the (m, X)th element of the matrix
N = ft'-rS ;
since the equations if = a + ra', Mr = /3 + rfi' give
-y8+Ta = (^-Ta)T,
we have also _
TOt.
z, c
These equations express the sum v% "'+... + C' °' in terms of integrals j
manner analogous to the expression originally taken for C '+...+ wt- '
in terms of integrals t;Ja,_the difference being the substitution, for the matrix
n a
386. The theory of correspondence of points of a Riemann surface now
divides into two parts according as the equation, which arises by elimination,
either of the matrix M or the matrix N, namely,
T«'T + O.T — T/3' — /3 = 0,
is true independently of the matrix r, in virtue of special values for the
matrices a, ft, a', /3', or, on the other hand, is true for more general values of
these matrices, in virtue of a special value for the matrix T.
We take the first possibility first ; it is manifest that for any value of r
the equation is satisfied if
o = -7#, ft = 0, «' = 0, ft' = -vE,
where 7 is any single integer, and E is the matrix unity of p rows and
columns ; conversely, if the equations are to hold independently of the value
of r, we must have the n2 equations
O, f a^r^
and, for general values of T, these give
0^ = 0, «m,« = /S/A,x, 9tiii+-fr&-*i A
which are equivalent to the results taken above.
386] EXISTING ON A GENERAL RIEMANN SURFACE. 643
With these values we have, as the particular forms of the general
equations of § 385,
Zi, C, Zr, Cr X, a ~
Vi + ...... + Vt +yVi =0,
x,,a, Xs,ag z,c f. /. -, -.
vnt + ...... +vm + yvm =0. (i, m=l, ...,p).
Let the value on the dissected surface of the left side of the first of these
equivalences be
gi + giTi,i + ...... + g 'P Ti, P >
where glt ..., gp, #/, . .., g'p are integers. Consider now the function
, Z]
Zr,C
+yll
z'c
wherein z1} ..., zr are the places corresponding to x, and clt ..., cr their
positions when x is at a, and z, c are arbitrary places. In virtue of the
equations just obtained it is a rational function of z, and rational in the
place c (cf. Chap. VIII. , § 158). Regarded as a function of x it is also
rational ; for the quotient of its values at the two sides of a period loop
of the second kind, which, by what has just been shewn, must be rational in
z, is, by the properties of the integral of the third kind, necessarily of the
form
where K1} ..., Kp are integers; this quotient, as a function of z, has no
infinities ; being a rational function of z, it is therefore a constant, and
therefore unity, since it reduces to unity when z is at c ; hence </> (x, z ; a, c),
as a function of x, has no factors at the period loops ; as it can have no
infinities but poles it is therefore a rational function of x; it is similarly
rational in a. As a function of x it vanishes when one of zl} ..., zr coincides
with z, that is, when x coincides with one of xlt .... xs.
We have therefore the result. Associated with any (r, s) correspondence
which can exist upon a perfectly general Riemann surface, it is possible to
construct a function $ (x, z; a, c), rational in the variable places x, z and the
fixed places a, c, which, regarded as a function of x vanishes to the first order
at the places xly ..., xg, which correspond to z, and vanishes to order 7 {if 7 be
positive), at the place z ; which, as a function of x, is infinite to the first order
when x coincides with any one of the places aly . .., ag which correspond to c,
and is infinite to order 7 (7 being positive) when x is at c; which, as a function
of z, has similarly (for 7 positive) the zeros zlt ..., zr, xt and the poles
Ci, ..., cr, a*. An analogous statement can be made when 7 is negative.
Ex. i. Some examples may be given to illustrate the form of this rational function.
On a plane cubic curve we do in fact obtain a (1, 4) correspondence, for which -y = 2,
by taking for the point zl which corresponds to .r, the point in which the tangent at
41—2
644 EXAMPLES OF GENERAL CORRESPONDENCES. [386
x meets the curve again, and therefore, for the points xv, x^ x3, x^ which correspond to 2,
the points of contact of tangents to the curve drawn from z. The value y = 2 is obtained
from Abel's theorem, which clearly gives the equation
z,, c, . n x, a -
v +2v =0
as representative of the fact that a straight line meets the curve twice at x and once at zt.
Denote the equation of the curve in the ordinary symbolical way by AX3 — Q ; then the
equation AXZA1 = 0, for a fixed position of x, represents the tangent at x ; and for a fixed
position of z, represents the polar conic of the point z, which vanishes once in the points of
contact, #!, #2, #3, #4, of tangents drawn from z and vanishes also twice at z, where it
touches the curve ; then consider the function
AX*A.
~A*At.A*An*
when z, a, c are fixed, this function of x vanishes to the first order at xlt #2, xs, x± and to
the second order at z, and is infinite to the first order at the places ax, a2, a3, a4 which
correspond to c, and infinite to the second order at c ; when z, a, c are fixed, this function
of z vanishes to the first order at zlt and to the second order at x, and is infinite to the
first order at the place Cj , which corresponds to a, and infinite to the second order at a.
Ex. ii. More generally for any plane curve of order n, and deficiency p, if to a point x
we make correspond the r = n — 2 points zl , . . . , zn _ 2 , in which the tangent at x meets the
curve again, and to a point z the s = 2n + 2p — 4 points of contact < ..., xt of tangents
drawn to the curve from z (so that, for instance, when the curve has K cusps, K of the
points a?j, ..., xt will be the same for all positions of z\ we shall have an (r, s) corre
spondence for which y = 2. If Aj.n = 0 be the equation of the curve, the function
regarded as a function of x, for fixed positions of z, a, c (of which a and c are not to be
multiple points), has for zeros the places xlt ..., xg, z2, for poles the places alt ..., a,, c2,
and regarded as a function of z, has for zeros the places zlt ..., zr, x2, and for poles the
places Cj, ..., cr, a2.
Ex. iii. If from a point x a tangent be drawn to a plane curve, and the corresponding
points be the points other than the point of contact, in which the tangent meets the curve
again, we have
where z1 is the point of contact of one of the tangents drawn from x, there being as many
such equations as tangents to the curve from x ; since the 2n + 2p — 4 points z1 lie on the
first polar of #, it follows by Abel's theorem that
2j
therefore
/"c' + ...... +
so that y = 2/i + 2jp-8. As a function of z the function <£ (#, z ; a, c) has therefore the
(n — 3) (2n + 2p — 4) zeros z1? ..., zr, which correspond to x, as well as the zero x, of the
(2n + 2p-8)th order, and has as poles the places cly ..., cr, which correspond to a, as well
as the zero a, of the (2n + 2p - 8)th order.
For instance for a plane quartic, there are 10 places corresponding to x, one for each of
the tangents that can be drawn from x to the curve ; the function $ (x, z ; a, c), as a
387] DETERMINATION OF THE COINCIDENCES. 645
function of z, vanishes to the first order at each of these ten places, and vanishes to the
sixth order at x ; its infinities are the places similarly derived from the fixed position, a,
of x. We can build up this function in the manner suggested by the use already made of
Abel's theorem in the determination of the value of y ; for a fixed position of x, let T(z) = 0
be the equation, in the variable 2, for the ten tangents to the quartic drawn from z ; let
P (z) = 0 be the first polar of x ; the quotient
vanishes when z is at the places zlt ..., z10, and vanishes when z is at x to order
10-2(2) = 6; let Ta(z\ Pa(z) represent what T(z\ P(z) become when x is at a ; then
the function of z
T(z)
has the same behaviour as has the function 0 (x, z ; a, c) as a function of z. From this
function, by multiplication by a factor involving x but independent of 2, we can form a
symmetrical expression in x and z ; this will be the function <f> (x, z ; a, c). In fact,
denoting the equation of the quartic curve by ^1^ = 0, and expressing the fact that the line
joining the point x of the curve to the point £ not on the curve should touch the curve,
viz., by equating to zero the discriminant in X of (Ax + \Atf-Ax*, we obtain an equation
of the form
^ [C6, *"] = (AXA£? [9 WAfF - IGAXA* . AJAJ,
which represents the tangents to the curve drawn from x. Replacing £ by z, a point on
the curve, so that A,* = 0, we have, since AXA ,3=0 is the first polar of x,
T(z)/P* W = 9(AX*A*)*-16AXA* . AJA. ;
hence
A(x z- a c) = XX..
[9 (Au*A*r - \<>AaA? . AJA.] [9 (AX*A*? - IQAXA* . Ax*Ae] '
Ex. iv. If a (1, 1) correspondence exists, the rational function of x, denoted by
<f> (x, z ; a, c), is of order y + 1.
387. A problem of great geometrical interest is to determine the number
of positions of x, in which x coincides with one of the places zl} ..., zr, which
correspond to it. This is called the number of coincidences.
A simple way to determine this number is to consider the rational func
tion of x obtained as the limit when z = a;,of the ratio $(x,z\ a, c)/(x — zf ;
putting
$(x\ a,c) = Km [<£ (x, z ; a, c)/(x - z)*],
and bearing in mind that if t be the infinitesimal on the Riemann surface,
dx/dt vanishes to the first order at every finite branch place, and is infinite to
the second order at every infinite place of the surface, we immediately find
from the properties of the function ^>(xtz\ a, c), on the hypothesis that none
of the branch places of the surface are at infinity, the following result ; the
rational function of x denoted by <j>(x; a, c) vanishes to the first order at
every place x of the surface at which x coincides with one of the places
646 DEDUCTION OF THE BITANGENTS [387
z1} ..., zr which correspond to it, vanishes also to order 2y at each of the n
infinite places of the surface, and is infinite to order 7 at each of the branch
places of the surface and at each of the places a, c, while it is infinite to the
first order at each of the places cx, ..., cr which correspond to a, and at each
of the places «1} ..., as which correspond to c ; hence, denoting the number of
coincidences by C we have
C + 2ny = (2n + 2p - 2) 7 + £7 + r + s,
so that*
The same result is obtained when there are branch places at infinity.
The argument has assumed 7 to be positive ; a similar argument, when 7 is
negative, leads to the same result.
Ex. i. The number, i, of inflexions of a plane curve of order n and deficiency p is
given (Ex. ii. § 386) by
where h is the number of coincidences arising other than inflexions, as for instance at the
multiple points of the curve. In determining h it must be remembered that we have not
excluded the possibility of there being fixed positions of x which correspond to z for all
positions of z ; for instance in the case of a curve with cusps all these cusps have been
reckoned among the places &\, ...,#„ which correspond to z. Therefore for a curve with
K cusps, h will contain a term 2« ; for a curve with only 8 double points and K cusps, the
formula is the well-known one
i— « = 3 (m — ri),
where m is the class of the curve, equal to n(n — l)-28 — 3x.
Ex. ii. Obtain the expression of the function <£ (x ; a, c) determined by the limit
{Af-*AJ(*-*?.Af-\A..AS-*A}^t
where Ax» = Q = A*=Aa» = Acn. (Cf. Ex. ii. § 386.)
Ex. iii. The number of double tangents of a curve of order n and deficiency p may be
obtained from Ex. iii. § 386, if we notice that a double tangent, touching at P and Q, will
arise both when P is a coincidence, and when Q is a coincidence ; hence if T be the number
of double tangents, and h the number of coincidences not giving rise to double tangents,
we have
where tr = n-\-p — 3. For instance for a -curve with no singular points other than 8 double
points and K cusps, there will be a contribution to h equal to twice the number of those
improper double tangents which are constituted by the tangents to the curve from the
cusps and the lines joining the cusps in pairs. The number of tangents, t, from a cusp is
given (cf. § 9, Chap. I., Ex.) by
-2, or i! = 2/i-5-
There will not arise any such contribution corresponding to a double point, since the two
* This result was first given by Cayley ; see, for references, Ex. iv. below.
387] OF AN ALGEBRAIC PLANE CURVE. 647
points of the curve that there correspond are different places (cf. § 2, Chap. I.) ; hence
we have
and therefore r = 2<r (a- + 1) - 4p - *t - 1(*2 - K) ;
substituting the values for <r, p and t, we find the ordinary formula equivalent to
where m is the class of the curve.
Ex. iv. The points of contact of the double tangents of a quartic curve AX4 = 0 lie
upon a curve whose equation is obtainable by determining the limit, when z=x, of the
expression
[9 ( Jz2 ^2)2 _ 16 AxA? . A*A^(X - Zf.
For the result, cf. Dersch, Math. Annal. vn. (1874), p. 497.
For the general geometrical theory the reader will consult geometrical treatises ; the
following references may be given here ; Clebsch-Lindemann-Benoist, Lemons sur la Geo
metric (Paris, 1879—1883), t. I. p. 261, t. n. p. 146, t. in. p. 76 ; Chasles, Compt. Rendus,
t. LVIII. (1864) ; Chasles, Compt. Rendus, t. LXII. (1866), p. 584 ; Cayley, Compt. Rendus,
t. LXII. (1866), p. 586, and London Math. Soc. Proc. t. I. (1865 — 6), and Phil. Trans.
CLVIII. (1868) (or Coll. Works, v. 542 ; vi. 9 ; vi. 263) ; Brill, Math. Annal. t. vi. (1873),
and t. vn. (1874). See also Lindemann, Crelle, LXXXIV. (1878); Bobek, Sitzber. d. Wiener
Akad., xcin. (ii. Abth.), (1886), p. 899 ; Brill, Math. Annal. xxxi. (1887), xxxvi. (1890) ;
Castelnuovo, Rend. Ace. d. Lincei, 1889; Zeuthen, Math. Annal. XL. (1892), and the
references there given.
Ex. v. If we use the equation (Chap. X. § 187)
where Q is an odd half-period, equal to X + rX' say, X, X' being each rows of p integers, and
form the rational function of x and a,
„. . 7. . _,y
R (x, a) = hmz=x ( - l)y — - , -
c=a 0 (#, * j 0, C)
[<t)(x, z; a, c
we have
l 1 IT,x, a
which is a generalisation of the equation (i), p. 427.
The function R(x, a) vanishes when x is at any one of the places c1? ..., cr, which
correspond to a, and when x is at any one of the places an ..., at which correspond to the
position a of the place c ; it vanishes also 2y times at each of the zeros of the function
e^.a-t-^Q). It is infinite C times, namely when x has any of the positions in which it
coincides with one of the places 2j, ..., zr which correspond to it. In the particular case
of Ex. i. p. 427, the function R(x,a) is (# - a)2 Jf (#), and the equation C=r + s + 2py
expresses that the number of branch places (where two places for which x is the same
coincide) is 2 (n- l) + 2p.
Ex. vi. Determine the periods of the function of x expressed by
648 EXISTENCE OF SPECIAL CORRESPONDENCES. [387
where zlt ..., zr are the places corresponding to x, and clt ..., cr are the places correspond
ing to a.
Ex. vii. If there be upon the same Riemann surface two correspondences, an (r, «)
correspondence and an (r', a') correspondence, then to any place z will correspond, in virtue
of the first correspondence, the places xlt ... , &•„ and to any one of these latter, say xit will
correspond, in virtue of the second correspondence, say z'itl, ..., z'i>r, ; conversely to any
place z' will correspond, in virtue of the second correspondence, the places xlt ..., xs/, and
to any one of these latter, say xt) will correspond, in virtue of the first correspondence,
say zitl, ..., zit,. ; we have therefore an (r's, rs') correspondence of the points (z, z'). In
virtue of the equations
we have
i=l J=l
Hence* we can make the inference. If upon the same Riemann surface there be two
correspondences, an (r, s) correspondence of places x, z, and an (/, s') correspondence of places
of, z1, then the number of common corresponding pairs of these two correspondences, for which
both x, x' coincide, and also z and z', is
r's + rs' — 2yy'p.
388. We have so far considered only those correspondences f which can
exist on any Riemann surface. We give now some results £ relating to
correspondences which can only exist on Riemann surfaces of special cha
racter, more particularly (1, 1) correspondences.
We prove first that any (1, s) correspondence is associated with equations
which are identical in form with those which have arisen in considering the
special transformation of theta functions. For any such correspondence, in
which to any place x corresponds the single place z, and to any position of z
the places xly ..., #„, we have shewn that we have the equations (i= 1, ...,£>)
vx;a, M=a +roc', Mr= ft+rff,
hence
£
' »P
z> c _ JUT 51 *«• ft» »*- •c' x>> a»
m=l
P
S MI &
A=i
r ^. c .
, T z>c
+Li>pvp ,
* Provided the (r's, rs') correspondence is not an identity.
t Called by Hurwitz, Werthigkeit-correspondenzen, y being the Werthigkeit.
{ For other results, see Klein-Fricke, Modulfunctionen, Bd. n. (Leipzig, 1892), pp. 540 if.
389] CONDITIONS FOR A (1, 1) CORRESPONDENCE. 649
where Li>m is the (i, m)th element of the matrix L, = MN. This matrix is
therefore equal to .9. Now
MN =M( J3'-Ta) = (a + ra) ft' - (/8 + r/3') a = aft'- /8o' + T(a'/8'-/8/a/),
MNr = M (- J3 + ra ) = - (a + ra) £ + (/3 + r/8') a = - (a)8 -/3o)+r(/3'a - a'^ ),
which we may write in the form
if now T = T1-MY2, where rlt T2 are matrices of real quantities, it follows
by equating to zero the imaginary part in the equation
that T2£ = 0; since for real values of ?i1} ..., np the quadratic form r.2n2 is
necessarily positive, the determinant of the matrix r2 is not zero ; hence we
must have B = 0 ; hence also H = s and A = 0 ; or
«£ = £*, <*'£'= /3'a', a/3'-/3a' = ffa-a.'/3 = s;
and these equations, with the equation (a + ra') r = 0 + r/3', are identical
in form with those already discussed in this chapter (§§ 377, ff.).
We are able then as in the former case to deduce certain conditions
for the matrices a, /3, a', /9', which in their general form necessarily involve
special values for the matrix T.
389. In particular, in order that a (1, 1) correspondence* may exist,
the roots of the equation M — A, = 0 must be conjugate imaginaries of the
roots of the equation \N— X j = 0, must be all of modulus unity, and must
be roots of the equation A — X ] = 0, where A = ( , ^, j . They must there-
\^ /"^ /
fore be roots of unity. For the sake of definiteness we shall suppose p > 1
and that A and r are such that the roots of | M — \ =0 are all different ;
this excludes the case already considered when A = ( /[ ) . Supposing
\ 0 — 7/
a (1, 1) correspondence to exist, for which this condition is satisfied, if in
the fundamental equations (i=l, ...,p)
z,c -, , x,a , Tif x,a
vt SMttl9l + ...... +Miipv1> ,
we introduce other integrals of the first kind, say Ff'", ..., F^'a, where
* The (1, 1) correspondence for the case p = l is considered in an elementary way in § 394.
The reader may prefer to consult that Article before reading the general investigation.
650 PERIODICITY OF A (1, 1) CORRESPONDENCE. [389
then we can put the fundamental equations into the form
for this it is necessary that X; should be a root of the equation M — \ = 0,
and that the p quantities ajl} ..., citp should be determined from the
equations
Ci,iM1>r+ ...... + citpMptr = \iCitr, (r= 1, ...,£>);
under the prescribed conditions the determinant of the matrix c will be
different from zero.
Hence as X^ is a root of unity, it can be shewn, when p> 1, that every
such (1,1) correspondence is periodic, with a finite period ; that is, if the place
corresponding to x be zlt the place corresponding to the position zl, of x,
be z2, the place corresponding to the position #2, of x, be z3, and so
on, then after a finite number of stages one of the places zl} z2, z3>...
coincides with x. In order to prove this, suppose that all the roots of the
equation | M — X \ = 0 are &-th roots of unity ; then denoting the place
x by z0 and the place a by c0, the equations of the correspondence may
be written
these give
and therefore
r 7 Zk,Ck j Zoic<n . r 7 z*> c* 7 zoip<n /\
Ci^ldv, -dvl ]+ ...... +Ci,p[dvp -dvp ] = 0;
hence on the dissected Riemann surface we have equations of the form
vr*' * - v?' ° = \r + X/Tr> ! + ...... + \p'rr> p> (r = l,...,p),
where X1( ..., Xp' are integers. Thus either zk = z0 and ck = c0, which is the
result we wish to obtain, or else there is a rational function expressed by
f-fX,a Ttx>a o -/\ / *i<* , , \ / x> a\
A.a-^.co-2^^^ + ...... + VV >,
which is of the second order, having zk, c0 as zeros and z0, ck as poles; now
a surface on which there is a rational function of the second order is
necessarily hyperelliptic (Chap. V. § 55) — but, on a hyperelliptic surface,
for which p>\, of the two poles of such a function either determines
the other, and of the two zeros either determines the other ; it is not
possible to construct such a function whereof, as here, one pole ck is fixed,
and the other arbitrary and variable (§ 52).
Hence we must have zk = z0, and ck = c0) which proves the result
enunciated.
There is no need to introduce the integrals V in order to establish this result. It
is known (Cayley, Coll. Works, Vol. n. p. 486) that if Xl5 X2, ... be the roots of the equation
|j^_A| = 0, the matrix M satisfies the equation (M— Xj) (M— X2) ...... = 0; when the roots
390] FORM OF THE FUNDAMENTAL ALGEBRAIC EQUATION. 651
Xj, X.j, ... are different &-th roots of unity it can thence be inferred that the matrix M
.satisfies the equation Mk = \ • then from the successive equations dv*1' c' = Mdv*0' c°,
dvz*' et=Mdvfl'Cl, etc., we can infer dvZk' °k=dvZl>' c°, and hence as before that zk=z0) ck=c0.
A proof of the periodicity of the (1, 1) correspondence, following different lines, and
not assuming that the roots of the equation \M—\\ = Q are different, is given by Hurwitz,
Math. Annal. xxxn. (1888), p. 295, for the cases when p>l. It will be seen below that
the cases p = 0, p = \ possess characteristics not arising for higher values of p (§ 394).
390. Assuming the periodicity of the (1, 1) correspondence, we can
shew that all Riemann surfaces upon which a (1, 1) correspondence exists,
can be associated with an algebraic equation of particular form. As before
let k be the index of the periodicity, and let w = eZnilk; let 8, T be any
two rational functions on the surface, and let the values of S at the
successive places x, zlt z.2, ..., zh_lt x which arise by the correspondence be
denoted by 8, Sl} ..., $A_1} S, and similarly for T ; then the values of the
functions
at the place zr are respectively
sr = S,. + w-] Sr+1 + ...... + o)-t*-i» Sr+M = a>rs, and t ;
hence it can be inferred (cf. Chap. I., § 4) that there exists a rational
relation connecting sk and t. Conversely S and T can be chosen of such
generality that any given values of s and t arise only at one place of
the original Riemann surface. Thus the surface can be associated with
an equation of the form
(«*,0 = 0,
wherein every power of s which enters is a multiple of k.
Such a surface is clearly capable of the periodic (1, 1) transformation
expressed by the equations
s' = MS, t' = t.
i* .
The following further remarkable results may be mentioned
(a) The index of periodicity k cannot be greater than 10 (p - 1).
(/3) When k > 2p - 2 the Riemann surface can be associated with an
equation of the form
«* = t*i (t - 1)*' (t - c)*».
(7) When k > 4>p - 4, the Riemann surface can be associated with an
equation of the form
sk = t^(t-l)^.
Herein klt Ar2, k3 are positive integers less than k.
* Hurwitz, Math. Annal. xxxn. (1888), p. 294.
652 UPPER LIMIT TO NUMBER OF COINCIDENCES. [391
391. We can deduce from § 389 that in the case of a (1, 1) correspond
ence the number of coincidences is not greater than 2p -f 2. In the case of
a hyperelliptic surface, when the correspondence is that in which conjugate
places — of the canonical surface of two sheets — are the corresponding pairs,
the coincidences are clearly the branch places, and their number is "2p 4- 2 ;
for all other (1, 1) correspondences on a hyperelliptic surface, the number of
coincidences cannot be greater than 4.
For, when the surface is not hyperelliptic, let g denote a rational function
which is infinite only at one place z0 of the surface, to an order p + 1 ; and
let g' be the value of the same function at the place z1} which corresponds
to z0 ; then the function g' — g is of order 2p + 2, being infinite to order
p + 1 at zQ and to order p + 1 at the place z^ to which z0 corresponds ; now
every coincidence of the correspondence is clearly a zero of g' — g ; thus
the number of coincidences is not greater than 2p + 2. In the case of a
hyperelliptic surface
2/2 = \x> 1)20+1 >
we may similarly consider the function x — x, of order 4 ; — unless the
correspondence be that given by y = — y, x' = x, for which x — x is identically
zero. We thus obtain the result that the number of coincidences cannot
be greater than 4, except for the (1, 1) correspondence y' = — y, x' = x.
It can be shewn for the most general possible (r, s) correspondence, associated with the
equations
by equating the value obtained for the following integral, taken round the period loops,
to the value obtained for the integral taken round the infinities of the subject of integra
tion, that the number of coincidences is
C=r+s-(an + + OPP+&II + +/3'pp)-
Since au + +/3V- ^s ^ne sum of the roots of the equation |A — X| = 0, it follows for a
(1, 1) correspondence, in which all the 2p roots of A — X|=0 are roots of unity, that
C^-2p + 2. For any (r, s) correspondence belonging to a matrix A=f ^ j, the same
formula gives C=r + s + 2py, as already found.
We have remarked (§ 386, Ex. iv.) for the case of a (1, 1) correspondence associated with
a matrix A of the form | l ) , the existence of a rational function of order 1 +y. For
\ (| ~yj
any such (1, 1) correspondence, if p be >1, y must be equal to +1 in order that the
number 1 + 1-f 2py of coincidences may be ;^>2jo + 2. Thus such a correspondence involves
the existence of a rational function of order 2, and involves therefore that the surface be
hyperelliptic. This is also obvious from the fact that such a correspondence is associated
with equations of the form
393] NUMBER OF (1, 1) CORRESPONDENCES IS LIMITED. 653
conversely, for y = 1, equations of this form are known to hold for any hyperelliptic surface,
associated with the correspondence of the conjugate places of the surface. From the
considerations here given, it follows for p>\ that for a (1, 1) correspondence the number
of coincidences can in no case be >2/? + 2.
392. In conclusion it is to be remarked that on any Riemann surface
for which p > 1, there cannot be an infinite number of (1, 1) correspondences.
For consider the places of the Riemann surface that can be the poles of
rational functions of order <(p+l) which have no other poles (§§ 28, 31,
34 — 36, Chap. III.). Denote these places momentarily as (/-places. As
such a (1, 1) correspondence is associated with a linear transformation of
integrals of the first kind, which does not affect the zeros of the de
terminant A, of § 31, it follows that the place corresponding to a ^-place
must also be a <jr-place. Now, when the surface is not hyperelliptic, every
#-place cannot be a coincidence of the correspondence; for we have shewn
(Chap. III., § 36) that then the number of distinct (/-places is greater
than 2p + 2; and we have shewn in this chapter (§ 391) that the number
of coincidences in a (1, 1) correspondence, when p>l, can in no case
be > 2p + 2. Therefore, when the surface is not hyperelliptic, a (1, 1)
correspondence must give rise to a permutation among the ^-places; since
the number of such permutations is finite, the number of (1, 1) corre
spondences must equally be finite. But the result is equally true for a
hyperelliptic surface; for we have shewn (§ 391) that for such a surface the
number of coincidences of a (1, 1) correspondence cannot be greater than 4,
except in the case of a particular one such correspondence; since the
number of distinct ^-places is 2p + 2, every (1, 1) correspondence other than
this particular one must give rise to a permutation of these ^-places. As
the number of such permutations is finite, the number of (1, 1) corre
spondences must equally be finite.
It is proved by Hurwitz* that the number of (1, 1) correspondences,
when p > 1, cannot be greater than 84 (p - I). In case p = 3, a surface is
known to exist having this number of (1, 1) correspond en cesf.
393. The preceding proof § (§ 392) is retained on account of its
ingenuity. It can however be replaced by a more elementary proof j by
means of the remark that a (1, 1) correspondence upon a Riemann surface
can be represented by a rational, reversible transformation of the equation of
the surface into itself. Let the equation of the surface be f(x,y) = Q;
let (z, s) be the place corresponding to (x, y) ; then z, s are each rational
functions of x and y such that f(z, s) = 0 ; conversely x, y are each
* Math. Annal. XLI. (1893), p. 424.
+ Klein, Math. Annal. xiv. (1879), p. 428; Modulfunctionen, t. i., 1890, p. 701.
§ Hurwitz, Math. Annal. XLI. (1893), p. 406.
£ Weierstrass, Math. Werke, Bd. 11. (Berlin, 1895), p. 241.
654 SELF-TRANSFORMATION OF A RIEMANN SURFACE. [393
rational functions of z, s. To give a formal demonstration we may
proceed as follows ; supposing the number of sheets of the Riemann surface
to be n, let z1} ..., zn denote the places corresponding to the n places
#1°', ,..,«£* for which x = 0, and let zj , ...,z'n denote the n places corre
sponding to the places d4 , ..., ae^ for which x is infinite ; as # is a rational
function on the surface we have, for suitable paths of integration (cf. Chap.
VIII. § 154)
J0> J«> JO) J«> - ,.
v*i'x> + ...... +'«J"f • =0, (i = l, ...,_p);
hence from the equations
z, c ir a;. <* . n f &• a
Vi = MitlVi + ...... +Miipvp ,
we have
there exists therefore (Chap. VIII., § 158) a rational function having the
places z-i, ..., zn as zeros, and the places zj, . .., zn' as poles ; regarding this as
a function of z, s and denoting it by (f> (z, s), it is clear therefore that x\$ (z, s)
is a constant, which may be taken to be 1. Hence x = (j>(z, s), etc.
For the theorem that for p>l the number of (1, 1) correspondences is limited the
reader may consult, Schwarz, Crelle, LXXXVII. (1879), p. 139, or Gesamm. Math. Abhand.,
Bd. II. (Berlin, 1890), p. 285 ; Hettner, Gotting. Nachr. (1880), p. 386 ; Noether, Math.
Annal., XX. (1882), p. 59 ; Poincare, after Klein, Acta Math., vn. (1885) ; Klein, Ueber
Riemann' s Theorie u. s. w. (Leipzig, 1882), p. 70 etc. ; Noether, Math. Annal., xxi. (1883),
p. 138 ; Weierstrass, Math. Werke, Bd. n. (Berlin, 1895), p. 241 ; Hurwitz, Math. Annal.,
XLI. (1893), p. 406.
394. In regard to the (1, 1) correspondence for the case p = l, some remarks may be
made. The case p = 0 needs no consideration here ; any (1,1) correspondence is expressible
by an equation of the form
thus there exists a triply infinite number of (1, 1) correspondences.
In case p = l, if there be a (1, 1) correspondence, whereby the variable place x
corresponds to #', and a, a! be simultaneous positions of x and x', it is immediately
shewn, if vF> a denote the normal integral of the first kind, that there exists an equation of
the form
tfV, a' = ptf, a?
wherein /* is a constant independent both of a and x. From this equation, by supposing x
to describe the period loops, we deduce eqxiations of the form
where a, a', /3, & are integers. By supposing x' to describe the period loops we deduce
equations of the form
'), (ii),
where y, y', 8, 8' are integers. The expression of these integers in terms of a, a', ft ft' is
394] ELEMENTARY TREATMENT OF THE ELLIPTIC CASE. 655
known from the general considerations of this chapter ; it is however interesting to
consider the equations independently. From the equations (ii) we deduce
8' - ry' = n (y8f - y'8), 8 - ry = - r/x (yS' - y'8) ;
if now y8'-y'8 = 0, either y and y are zero, which is inconsistent with 1 =p (y + ry'), or else
T is a rational fraction ; it is known that in that case the deficiency of the surface is not 1
but 0 ; we may therefore exclude that case ; if y8' - y'8 be not zero, we have
hence, unless T be a rational fraction, we have
*' ' &
° -y _, y _ff -o _0
yV-y'6 ' y8'-y'8~ ' y8'-y'8~P' y8'-y'8~P>
and therefore
l = (a/3'-a'/3)(y8'-y'8);
thus af? — a'P = y8' -y'8 = + 1 or - 1 ; let t denote their common value ; then we deduce
fc/ f f £\t & n
a = ea, y = — a e, y = pt, o= — pe ;
by these the equations (ii) lead to
that is, to the equations (i).
Further, from the equations (i) we deduce in turn
so that /i is a root of the equation
-/* ft =0;
a /3' — u
now if a be zero, the first of equations (i) gives p = a, and, therefore, as r cannot be
the rational fraction 0/(a-00, the second of equations (i) gives a=p', 0=0 ; the equations
give /*2=f, or, since /u, =a, is an integer, they require e= +1 and /i=+l or/i=-l; the
equations corresponding to /* = + 1 and p = — I are
these do belong to existing correspondences — of the kind considered in §§ 386, 387, the
coefficient y being ±\*. But they differ from the (1,1) correspondences which are possible
whenp>l, in each containing an arbitrary parameter ;
if next, a' be not zero, the equation for T gives
2ra' = - (a - 00 ± \i(a + pj-4f,
so that, as T cannot be real, we must have
(« + 0')2-4f«),
* For instance, on a plane cubic curve, the former equation is that in which to a point of
argument u we make correspond the point of argument u + constant ; the line joining these two
points envelopes a curve of the sixth class, which in case the difference of arguments be a
half-period becomes the Cayleyan, doubled ; while the latter equation is that in which we
make correspond the two variable intersections of a variable straight line passing through a
fixed point of the cubic.
656 THE ELLIPTIC CASE. [394
and this shews that, in this case also, e = l. Hence the equations are reduced to precisely
the same form as those already considered for the special transformation of theta functions
(§ 383) ; and the result is that the only special surfaces, having p = l, for which there exists
a (1, 1) correspondence are those which may be associated with one of the two equations
the former has the obvious (1, 1) correspondence given by x' = —A; y' = iy ; the latter has
the obvious correspondence given by x1 = e 3 x, /i/=y ; the index of periodicity is 2 in the
former case and 3 in the latter case.
Ex. Consider the (1, 2) correspondence on a surface for which p = \ in a similar way.
For the equation
7/2 = 8^-15^ + 7
shew that a (1, 2) correspondence is given (cf. Ex. ii. § 383) by
8 (.*•-!)' * * 4 (x-\f
395]
CHAPTER XXII.
DEGENERATE ABELIAN INTEGRALS.
395. THE present chapter contains references to parts of the existing
literature dealing with an interesting application of the theory of trans
formation of theta functions.
It was remarked by Jacobi* for the case p = 2, that if the fundamental
algebraic equation be of the form
2/2 = x (x - 1) (x — K) (x — \) (x — K\),
an hyperelliptic integral of the first kind is reducible to elliptic integrals ;
in fact, putting | = x + K\/X, we immediately verify that
_ (x ± V/cX) dx _ __ df _
-«X) </(£+ 2 V/t
396. Suppose more generally that for any value of p there exists an
integral of the first kind
U = X^^ 4- ...... -f \pUp,
wherein ul}...,up denote the normal integrals of the first kind, which is
reducible to the form
_
R(%) being a cubic polynomial in f, such that £ and (f) are rational
functions on the original Riemann surface; then there exist p pairs of
equations of the form
wherein at-, &;, a/, &/ are integers ; we may suppose H' to be chosen so that
the *2p integers
a1} ..., ap, a/, ..., ap'
have no common factor and so that
aM + a2b2' + ...... + ctpbp — a/tj - a/62 — ...... —ap'bp = r,
* Crelle, vm. (1832), p. 41G.
B. 42
658 TRANSCENDENTAL CONDITIONS [396
where r is a positive integer; we assume that r is not zero. Eliminating
the quantities X1; ..., \p, and putting &> = H'/Il, we have the p equations
if therefore the matrix of integers, A = f , \L, ] , of 2p rows and columns,
\(* ^j /
wherein the first column consists of the integers aly ..., ap' in order, and the
(p + l)th column consists of the integers bl} ..., bpr in order, be determined
to satisfy the conditions for a transformation of order r,
oa' = a a, J30' = ff&, a/3' - a/3 = r,
(§ 420, Appendix II.), then it immediately follows from the equation, for
the transformed period matrix T', namely
that r'u = to, r'12 = 0, ..., r'y, = 0 ; to see this it is sufficient to compare the
elements of the first columns of the two matrices /3 + r/3', (a + rot)r'. In
other words, when there exists such a degenerate integral of the first kind as
here supposed, it is possible*, by a transformation of order r, to arrive at
periods r for which the theta function ^(w, T' \ q) is a product of an elliptic
theta function, in the variable wlt and a theta function of (p— 1) variables,
w2, ...,wp.
397. It can however be shewn that in the same case it is possible by a
linear transformation to arrive at a period matrix r" for which
r"13=0, T*14 = 0, ...,r% = 0,
while r"i2, = 1/r, is a rational number. We shall suppose -f* two rows oc, x .
each of p integers, to be determined satisfying the equations
ax — ax = 1, bx — b'x = 0,
such that the 2p elements of rx — b, rx' — b' have unity as their greatest
common factor, a denoting the row a1} ..., ap, etc., and suppose (§ 420) a
matrix of integers, of 2p rows and columns,
x, ..
» ** — / / 11
6J \a, rx -b, ...
to be determined, satisfying the conditions for a linear transformation,
^y=ry'ry) SB' =8' 8, j8' — j8 = l,
wherein the first column consists of the elements of a and a', the second
column consists of the elements of rx — b and rx —b', and the (jp + l)th
* This theorem is due to Weierstrass, see Konigsberger, Crelle, LXVII. (1867), p. 73 ; Kowal-
evski, Acta Math. iv. (1884), p. 395. See also Abel, (Euvres, t. i. (1881), p. 519.
t The proof that this is possible is given in Appendix II., § 419. It may be necessary, before
hand, to make a linear transformation of the periods ft, ft'.
398] FOR THE EXISTENCE OF A DEGENERATE INTEGRAL. 659
column consists of the elements of as and x ; the conditions for a linear trans
formation, so far as they affect these three columns only, are
a (rx' _ &') _ a' (rx - b) = 0, ax -a'x=l, (rx -b)x - (rx -b')x = Q,
and these are satisfied in virtue of the equation ab' — a'b = r. Then the
equation for the transformed period matrix r", namely
(7 + Ty') T" = 8 + T&,
leads to T"S, i = 0, . . . , T"PI t = 0 if only the p equations
[7<,i + (T7 ki] A, + [7,-|8 + (T7')i,J T7/a>1 = Bitl + (rS'Xi, (i = 1, ..-, P),
which are obtained by equating corresponding elements of the first columns
of the matrices S + rS', (7 + T7')T", are satisfied; these p equations are
included in the single equation
T"J, i [a + TO,'] + T\ i[ras-b + r (rx - b')] = x + rx,
and are satisfied* by T"V = ta/r, T"2)1 = l/r ; for we have, as the fundamental
condition, the equation
a) (a + TO!) = b + rb'.
398. It follows therefore in case p= 2 that the matrix r" has the form
"u, 1M .
/r, r"J '
hence it immediately follows that beside the integral of the first kind already
considered, which is expressible as an elliptic integral, there is another
having the same property. In virtue of the equations here obtained the first
integral having this property can be represented, after division by fi, in the
form
U = (V -rr\ XK
where u denotes the row of 2 integrals uly u2 ; consider now the integral
V = [rf - a' - rr"2>2 (rx - b')] u,
where t' is a row of two elements, these being the constituents of the first
column of the matrix 8'; the periods of Vat the first set of period loops are
given by the row of quantities
rtf — a — ?'T"2)2 (rx' — b'),
* See Kowalevski, Acta Math. iv. (1884), p. 400 ; Picard, Bulletin de la Soc. Math, de France,
t. xi. (1882—3), p. 25, and Conipt. Itendtis, xcn. xciu. (1881); PoincarS, Bulletin de la Soc. Math,
de France, t. xn. (1883—4), p. 124 ; Poincare, American Journal, vol. vin. (1886), p. 289.
42—2
660 CONNEXION WITH THEORY OF COMPLEX MULTIPLICATION. [398
and are linear functions of the two quantities 1, rr"2i2; the periods of Fat
the second set of period loops are given by
[r (rt' - a')l - rr"2,2 [T (rx' - V)]it (» = 1, 2) ;
now the equation (7 + TJ) r" = 8 + r8' gives
(7 + T7)f,i r\2 + (y + ry\, r"2,2 = (8 + rS\2, (i = 1, 2),
and hence we have
T*I,S [a + ra] + r"2>2 [rx-b + r (rx - b')] = t + rt',
where t is the row formed by the constituents of the first column of the
matrix B; therefore, as T//1)2=l/r, the periods of V at the second set of
period loops are expressible in the form
- (rt - a)i + rr\t 2 (rx - b){ , (i = 1 , 2),
and these are also linear functions of the two quantities 1, rr"2j2. Hence it
may be inferred that the integral V is reducible to an elliptic integral.
399. It has been shewn in the last chapter that for special values of the
periods T there exist transformations of the theta functions into theta func
tions for which the transformed periods are equal to the original periods. It
can be shewn* that for the special case now under consideration such a
transformation holds. Suppose that a theta function S-, with period r, is
transformed, as described above, into a theta function <£, with period T, for
which r'li2= 0 = ... = T'I>P, by a transformation associated with the matrix
A = ( , ni } 5 suppose further that there exists, associated with a matrix
H — ( „ , , ) , a transformation whereby the theta function <f> is transformed
\x nJ
into another theta function with the same period T' ; then it is easy to prove
that there exists a corresponding transformation of the theta function ^
whereby it becomes changed into a theta function with the same period T,
namely the transformation is that associated with the matrix
,f g'J U/3'AxVA-a «
to prove this it is only necessary to shew that the equations
(X + r'A/) r' = fj, + r'/jf, (a + rot) T' = {3 + r/3'
give the equation
Wiltheiss, Math. Annal. xxvi. (1886), p. 127.
401] ALGEBRAIC CONDITIONS FOR A DEGENERATE INTEGRAL. 661
Hence it follows that in order to determine a transformation of the function
^ which leaves the period r unaltered, it is sufficient to determine a trans
formation of the function <f> which leaves the period r unaltered ; this
determination is facilitated by the special values of T\,Z, ..., T\,P\ and in
fact we immediately verify that the equation (A, + T'\') T' = //- + T'JA is satisfied
by taking X' = p = 0 and by taking each of X and p' to be the matrix in
which every element is zero except the elements in the diagonal, each of
these elements being 1 except the first, which is — 1.
400. Thus for the case p = 2, supposing r = 2, the original function ^ is
transformed into a theta function with unaltered period T, by means of the
transformation of order 4 associated with the matrix,
where m denotes the matrix ( j ; the matrix V is equal to 2A"1, and it
is easy to see that this transformation of order 4 is equivalent to a multipli
cation, with multiplier 2, together with a linear transformation associated
with the matrix
We have therefore the result ; when, in case p — 2, there exists a transforma
tion of the second order whereby the periods r are changed into periods T' for
which T'IF 2 = 0, then there exists a linear transformation whereby the periods
T are changed into the same periods r, or what we have called in the last
chapter a complex multiplication.
401. The transcendental results thus obtained enable us to specify the
algebraic conditions for the existence of an integral of the first kind which is
reducible to an elliptic integral.
Thus for instance when p = 2, to determine all the cases in which an
integral of the first kind can be reduced to an elliptic integral by means of a
transformation of the second order, A = (, Q, } , it is sufficient to consider
\a p )
the conditions that the transformed even theta function ^\w\ T' ^L ])
may vanish for zero values of w ; for when T',i2 = 0 this function breaks up into
the product of two odd elliptic theta functions. By means of the formulae*
for transformation of the second order, it can be shewn*f- that this condition
leads to the equation
* Chap. XX. § 364.
t Konigsberger, Crelle, LXVII. (1867), p. 77.
662 ALGEBRAIC CONDITIONS. [401
and by means of the relations expressing the constants of the fundamental
algebraic equation in terms of the zero values of the even theta functions* it
can be shewn that this is equivalent to the condition that the fundamental
algebraic equation may be taken to be of the form
2/2 = x (x — 1) (x — K) (x — \)(x — K\),
so that the case obtained by Jacobi is the only one possible for transformations
of the second order.
In the same case of p = 2, r = 2, the same result follows more easily from
the existence, deduced above, of a complex multiplication belonging to a
transformation of the first order. For it follows from this fact that the
algebraic equation can be taken in a form in which it can be transformed
into itself by a transformation in which the independent variable is trans
formed by an equation of the form
_
~
and this leadsf to the form, for the fundamental algebraical equation,
S2 = (02 _ ft2) ^a _ fc) ^ _ ca^
which is immediately identified with the form above by putting
X = \/KX(Z +
the quantities a, b, c being respectively
Similarly for p = 3, when the surface is not hyperelliptic, it can be shewn j
from the relations connecting the theta functions when a theta function is the
product of an elliptic theta function and a theta function of two variables,
that the only cases in which an integral of the first kind can be reduced to
an elliptic integral are those in which the fundamental algebraic equation
can be taken to be of the form
Jx(Ax+By) + \/y(Cx + Dy) + Vl + Fas + Gy = 0.
The Riemann surface associated with this equation possesses a (1, 1) corre
spondence given by the equations
* Cf. Ex. v. p. 341. By means of the substitution x = cl + (a1-c1)l-, the branch places can be
taken at £ = 0, 1, K, X, /x, wherein, if cx, a1( c2, a2, c be real and in ascending order, 0, 1, K, X, ft
are in ascending order of magnitude. For complete formulae, when the theta functions are
regarded as primary, and the algebraic equation as derived, see Eosenhain, Mem. p. divers
Savants, xi. (1851), p. 416 ff.
t Wiltheiss, Math. Annal. xxvi. (1886), p. 134.
£ Kowalevski, Acta Math. iv. (1884), p. 403.
4-03] REFERENCES. 663
402. But the problem of determining the algebraic equations for which an associated
integral of the first kind reduces to an elliptic integral may be considered algebraically, by
beginning with an elliptic integral and transforming it into an Abelian integral. The
reader may consult Richelot, Crelle, xvi. (1837); Malet, Crelle, LXXVI. (1873), p. 97;
Brioschi, Compt. Rendus, LXXXV. (1877), p. 708; Goursat, Bulletin de la Soc. Math, de
France, t. xni. (1885), p. 143, and Compt. Rendus, c. (1885), p. 622 ; Burnside, Proc. Lond.
Math. Soc. vol. xxm. (1892), p. 173.
403. The paper of Konigsberger already referred to (Crelle, LXVII.) deals with the case
of a transformation of the second order, for p = 2. For the case of a transformation of the
third order, when p = 2, consult, beside the papers of Goursat (loc. cit. § 402), also
Hermite, Ann. de la Soc. Scient. de Bnixelles, 1876, and Burkhardt, Math. Annal. xxxvi.
(1890), p. 410. For the case p = 2, and a transformation of the fourth order, see Bolza,
Ueber die Reduction hyperelliptischer Integrale u. s. w., Getting. Dissertation (Berlin,
Schade, 1885), or Sitzungsber. der Naturforsch. Ges. zu Freiburg (1885). The paper of
Kowalevski (Ada Math, iv.) deals with the case of a transformation of the second order for
p = 3. See further the references given in this chapter, and Poincare, Compt. Rendus,
t. xcix. (1884), p. 853 ; Biermann, Sitzungsber. der Wiener Akad. Bd. LXXXVII. (ii. Abth.)
(1883), p. 983.
[404
APPENDIX I.
ON ALGEBRAIC CURVES IN SPACE.
404. GIVEN an algebraic curve ((7) in space, let a point 0 be found, not on the curve,
such that the number of chords of the curve that pass through 0 is finite ; let the curve
be projected from 0 on to any arbitrary plane, into the plane curve (/), and referred to
homogeneous coordinates £, r], T in that plane, whose triangle of reference has such a
position that the curve does not pass through the angular point TJ, and has no multiple
points on the line r=0; let the curve (C) be referred to homogeneous coordinates £, 77, f, T
of which the vertex f of the tetrahedron of reference is at 0. Putting X = £/T, y = r)/r,
Z = C/T, it is sufficient to think of x, y, z as Cartesian coordinates, the point 0 being at
infinity. Thus the plane curve (/) is such that y is not infinite for any finite value of x,
and its equation is of the form f(y, x)=ym + A1ym~1 + ...... +Am=Q, where A^..,,Am
are integral polynomials in x ; the curve (C) is then of order ra; we define its deficiency
to be the deficiency of (/); to any point (x, y) of (/) corresponds in general only one
point (x, y, z) of (C), and, on the curve (C), z is not infinite for any finite values of x, y.
Now let /' (y) = 9/(y, x)fiy, let <£ be an integral polynomial in x and y, so chosen
that at every finite point of (/) at which f'(y) = 0, say at x=a, y = b, the ratio
(x - a) (fr/f (y) vanishes to the first order at least ; let a = n (x- a) contain a simple factor
corresponding to every finite value of x for which /'(y) = 0; let y±, ...,ym be the values
of y which, on the curve (/), belong to a general value of x, so that to each pair (x, y>)
there belongs, on the curve (C), only one value of z; considering the summation
% (c-yQ ...... (c-ym)r ^ 1
i=i c-Vi L/' (#)>="*'
where c is an arbitrary quantity, we immediately prove, as in § 89, Chap. VI., that it
has a value of the form
a (cm~l ttx + cm~2 U2 + ...... + wj,
where «1,...,«m are integral polynomials in x\ putting yi for c, after division by a, we
therefore infer that z can be represented in the form
where 0, ^ are integral polynomials in x and y, whereof </> is arbitrary, save for the
conditions for the fractions (x - a] <£//' (y). This is Cayley's monoidal expression of a
curve in space with the adjunction of the theorem, described by Cay ley as the capital
theorem of Halphen, relating to the arbitrariness of 0 (Cayley, Collect. Works, Vol. v. 1892,
p. 614).
404] ALGEBRAIC CURVES IN SPACE. 665
It appears therefore that a curve in space may be regarded as arising as an
interpretation of the relations connecting three rational functions on a Riemann surface ;
and, within a finite neighbourhood of any point of the curve in space, the coordinates
of the points of the curve may be given by series of integral powers of a single quantity t,
this being the quantity we have called the infinitesimal for a Riemann surface; to
represent the whole curve only a finite number of different infinitesimals is necessary.
More generally the representation by means of automorphic functions holds equally well
for curves in space. And the theory of Abelian integrals can be developed for a curve
in space precisely as for a plane curve, or can be deduced from the latter case; the
identity of the deficiency for the curve in space and the plane curve may be regarded as
a corollary. Also we can deduce the theorem that, of the intersections with a curve in
space of a variable surface, not all can be arbitrarily assigned, the number of those whose
positions are determined by the others being, for a surface of sufficiently high order, equal
to the deficiency of the curve.
Ex. If through p - 1 of the generators of a quadric surface, of the same system, a
surface of order p + l be drawn, the remaining curve of intersection is representable by
two equations of the form
y = W 1 /2P + 2 > ZU1 = M2 '
where (x, l)2P + 2 is an integral polynomial in x of order 2/? + 2, and wn w2 are respectively
linear and quadric polynomials in x and y.
For the development of the theory consult, especially, Noether, Abh. der Akad. zit
Berlin vom Jahre 1882, pp. 1 to 120 ; Halphen, Journ. £cole Polyt., Cah. LII. (1882),
pp. 1—200; Valentiner, Acta Math., t. n. (1883), pp. 136—230. See also, Schubert,
Math. Annal. xxvi. (1885); Castelnuovo, Rendiconti delta R. Accad. dei Lincei, 1889;
Hilbert, Math. Annal., xxxvr. (1890).
[405
APPENDIX II.
ON MATRICES*.
405. A SET of n quantities
(#j , . . . , Xn)
is often denoted by a single letter a, which is then called a row letter, or a column letter.
By the sum (or difference) of two such rows, of the same number of elements, is then
meant the row whose elements are the sums (or differences) of the corresponding elements
of the constituent rows. If m be a single quantity, the row letter mx denotes the row
whose elements are mx^ ..., mxn. If x, y be rows, each of n quantities, the symbol xy
denotes the quantity xlyl -f- + xnyn.
406. The set of n equations denoted by
#< = 0i,i!i + + ai,p£j» (i=l, ,n)
where n may be greater or less than p, can be represented in the form a;=a£, where a
denotes a rectangular block of np quantities, consisting of n rows each of p quantities,
the r-th quantity of the i-ih row being ai>r. Such a block of quantities is called a
matrix ', we call ait r the (i, r)th element of the matrix. The sum (or difference) of two
matrices, of the same number of rows and columns, is the matrix formed by adding (or
subtracting) the corresponding elements of the component matrices. Two matrices are
equal only when all their elements are equal ; a matrix vanishes only when all its
elements are zero. If £x , . . , £p be expressible by m quantities X1,...) Xm by the equations
f. = &r,i^l + +1>r,mXm* (r=l, 2> ,P\
so that £ = bX, where b is a matrix of p rows and m columns, then we have
•^ = ct,i^i + +ci>mXm, (i=l, , n),
or x=cX, where
/i=l, , n\
c;,s=«i, 101,8 + +ai,pbp>s, I I,
\° — i) > m/
* The literature of the theory of matrices, or, under a slightly different aspect, the theory of
bilinear forms, is very wide. The following references may be given : Cayley, Phil. Trans. 1858,
or Collected Works, vol. n. (1889), p. 475 ; Cayley, Crelle, L. (1855) ; Hermite, Crelle, XLVII.
(1854) ; Christoffel, Crelle, LXIII. (1864) and LXVIII. (1868) ; Kronecker, Crelle, LXVUI. (1868) or
Gesam. Werke, Bd. i. (1895), p. 143 ; Schlafli, Crelle, LXV. (1866) ; Hermite, Crelle, LXXVIII.
(1874) ; Kosanes, Crelle, LXXX. (1875) ; Bachmann, Crelle, LXXVI. (1873) ; Kronecker, Berl.
Monatsber., 1874; Stickelberger, Crelle, LXXXVI. (1879); Frobenius, Crelle, LXXXIV. (1878),
LXXXVI. (1879), LXXXVIII. (1880) ; H. J. S. Smith, Phil. Trans., CLI. (1861), also, Proc. Lond. Math.
Soc., 1873, pp. 236, 241 ; Laguerre, J. d. Vec. Poly.,t. xxv., cah. XLII. (1867), p. 215 ; Stickelberger,
Progr. poly. Schule, Zurich, 1877 ; Weierstrass, Berl. Monats. 1858, 1868 ; Brioschi, Liouville,
xix. (1854) ; Jordan, Compt. Rendus, 1871, p. 787, and Liouville, 1874, p. 35 ; Darboux, Liouville,
1874, p. 347.
408] INTRODUCTORY ACCOUNT OF MATRICES. 667
cit g being the (i, .<)th element of a matrix of n rows and m columns ; it arises from the
equations x=a£, £ = &Jf, whereof the result may be written x=abX ; hence we may
formulate the rule : A matrix a may be multiplied into another matrix b provided the
number of columns of a be the same as the number of rows of b ; the (i, s)th element of the
resulting matrix is the result of multiplying, in accordance with the rule given above, the
\-th row of a by the &-th column of b. Thus, for multiplication, matrices are not generally
commutative, but, as is easy to see, they are associative.
The matrix whose (i, «)th element is cg>i, where cSii is the (s, ?')th element of any
matrix c of n rows and m columns, is called the transposed matrix of c, and may be
denoted by c ; it has m rows and n columns, and, briefly, is obtained by interchanging the
rows and columns of c. The matrix which is the transposed of a product of matrices is
obtained by taking the factor matrices in the reverse order, each transposed ; for example,
if a, 6, c be matrices,
abc=cba.
407. The matrices which most commonly occur are square matrices, having an equal
number of rows and columns. With such a matrix is associated a determinant, whose
elements are the elements of the matrix. When the determinant of a matrix, a, of p rows
and columns, does not vanish, the p linear equations expressed by x = a% enable us to
represent the quantities £t, ..., £p in terms of a,\, ..., xv ; the result is written £ = a~lx, and
a"1 is called the inverse matrix of a ; the (i, r)th element of a~l is the minor of ar< f in
the determinant of the matrix a, divided by this determinant itself. The inverse of a
product of square matrices is obtained by taking the inverses of the factor matrices in
reverse order ; for example, if a, b, c be square matrices, of the same number of rows and
columns, for each of which the determinant is not zero, we have
The inverse of the transposed of a matrix is the transposed of its inverse ; thus
The determinant of a matrix a being represented by | a \ , we clearly have | ah \ = \ a \ \b\.
408. Finally, the following results are of frequent application in this volume : (i) If a
be a matrix of n rows and p columns, and £ a row of p quantities, the symbol a£ denotes
a row of ?i quantities ; if TJ be a row of n quantities, the product of these two rows, or
(^X7?)* is denoted by agij. When n=p this must be distinguished from the matrix
which would be denoted by a . fr — this latter never occurs. We have then
and this is called a bilinear form ; we also clearly have the noticeable equation
(ii) if b be a matrix of n rows and q columns, the product of the two rows «£, brj, wherein
is now a row of q quantities, is given by either (ba) £17 or (ab) qg, so that we have
The result of multiplying any square matrix, of p rows and columns, by the matrix Et
of p rows and columns, wherein all the elements are zero except the diagonal elements,
which are each unity, is to leave the multiplied matrix unaltered. For this reason the
matrix E is often denoted simply by 1, and called the matrix unity of p rows and
columns.
668
SYSTEMS OF UNITIES FORMED WITH MATRICES.
[409
409. Ex. i. If a bilinear form axy, wherein x, y are rows of p quantities, and a is a
square matrix of p rows and columns, be transformed into itself by the linear substitution
x= R£, y = *S'»7, where R, S are matrices of p rows and columns, then aR£. iSr) = a^rj ; hence
SaR = a.
Ex. ii. If h be an arbitrary matrix of p rows and columns, such that the determinants
of the matrices a + h do not vanish, and the determinant of the matrix a do not vanish,
prove that
(a+h}a~l (a-h)=a-ha~1 h = (a — h)a~1 (a+h) ;
hence shew that if
R = a~1(a-h)(a + h')-1a, S=a (a-h)~l (a + h)a~\
the substitutions x = R%, y — Sr] transform axy into a^rj.
For a substitution in which R = S see Cayley, Collected Works, vol. n. p. 505. Cf. also
Taber, Amer. Journ., vol. xvi. (1894) and Proc. Lond. Math. Soc., vol. xxvi. (1895).
Ex. iii. The matrices, of two rows and columns,
P (l °\ 7
Mo I/I "
give E2 = jE, J2= — E ; and the determinant of the matrix
vanishes, for real values of x, y, only when x — 0, y = 0.
Ex. iv. The matrices, of four rows and columns,
/I 0 0 0\ / 0 1 0
e = l
°
0 010
0 001
-1 000
0-100'
(oo o r
00-10
01 00
-10 00'
00107 \ 000 -II
10 0 0 V \ 0 0 1 0
give j*=jf=j3*=-e, jJ3= -J3j2=ji, J3Ji=-JiJ3=J2,
Hence these matrices obey the laws of the fundamental unities of the quaternion
analysis. Further the determinant of the matrix
which is equal to (x^+^+x^+a;^, vanishes, for real values of x, x^ x2, x3, only when
each of a?, tclt #2, xz is zero. (Frobenius, Crelle, LXXXIV. (1878), p. 62.)
410. In the course of this volume we are often concerned with matrices of 2p rows
and 2p columns. Such a matrix may be represented in the form
/* =
wherein a, b, c, d are square matrices with p rows and columns ; if // be another such
matrix given by
b'
411] REDUCTION OF GENERAL ABELIAN MATRIX. 669
the (i, r)th element of the product /*'/*, when i and r are both less than p + 1 is
a'ft j <Z1) r -f- + Q> i , p dp, r 4" b'i, \ C\, r + -f- ^'i, p £/>, r >
and this is the sum of the (i, r)ih elements of the matrices a'a, b'c ; similarly when i and r
are not both less than p + 1 ; hence we may write
fa' 6'\ fa 6\ _ fa! a + b'c , a'b + 6'cA
\c' d'J \c d) ~\c'a+d'c, c'b + d'd)'
the law of formation for the product matrix being the same as if a, b, c, d, a', b', c', d' were
single quantities.
Ex. Denoting the matrices ( n 1 ) , ( , A ] respectively by 1 and j, the matrices of
0 -
,
Ex. iv. can be denoted by
/I 0\ f-jO\ ( 0 IN / 0
e=(oi)> *~( oy> *K-io)' *-(-,-
411. We proceed now to prove the proposition* assumed in § 333, Chap. XVIII.
Retaining the definitions of the matrices Ak, B, C, D there given, and denoting
Ak~l, B~l, C~l, D'1 respectively by ak, b, c, d, we find
and
= Ak, so that Ak2 = l,
b = ( 01 ), c = ( 1 1 ), d=( 1 01 )
1 0
1 0
1 10
1 0
1 0
1 0
1 0
1 0
1 0
-1 0
0 1
0 1
0 1
0 1
0 1
0 1
0 1
0 1
0 1
0 1
0 1
so that b, c, d differ respectively from B, C, D only in the change of the sign of the
elements which are not in the diagonal. It is easy moreover to verify such facts as the
following
which are equivalent respectively with
64=1, (cb)3=l, a2d=da2, bakbak = akbakb,
but such results are immediately obvious from the interpretations of the matrices ak, b, c, d
which are now to be given.
Let A denote any matrix of 2p rows and columns, and let the four products
A''; . -V'. AC, _V/
* For a shorter proof of an equivalent result the reader may consult C. Jordan, Traite des
Substitutions (Paris, 1870), p. 174. The theorem was first given by Kronecker, " Ueber bilineare
Formen," Monatsber. Berl. Akad. 1866, Crelle, LXVIII. or in Werke (Leipzig, 1895), Bd. i. p. 160 ;
the proof here given follows the lines there indicated.
670 REDUCTION OF GENERAL [411
be formed ; the resulting matrices will differ from A in respects which are specified in the
following statements :
(i) ak interchanges the first and Mh columns (of A), and, at the same time, the
(p + l)th and (p + k)th columns (l< k< p + l). For the sake of uniformity we introduce
also al5 =1.
(ii) b interchanges the first and (^> + l)th columns, at the same time changing the
signs of the elements of the new first column.
(iii) c adds the first column to the (jt) + l)th.
(iv) d adds the first and second columns respectively to the (p + 2)ih and
the Qo + l)th.
Hence we have these results : if the matrices denoted by the following symbols be
placed at the right side of any matrix A, of 2p rows and columns, so that the matrix
A acts upon them, the results mentioned will accrue : —
Ik=akb2ak, changes the signs of the k-th and (p + k)th columns (of A),
t}c = akbalc, interchanges the k-th and (p + k}th columns (of A), giving the new k-th
column an opposite sign to that it had before its change of place,
t'k = akb5ak, interchanges the k-th and (p + k)th columns, giving the new (p + k)th
column a changed sign.
mk=akb2cb*ak, adds the k-th column to the (p + k)th.
m'ic=akb3cbcb3ak=aicb'2c~1b2ak, subtracts the k-th column from the (p + k)th.
nk = akb2cbcak = akbc~1b3ak, adds the (p + k}th column to the k-th.
n'k=akb3cbak, subtracts the (p + k)th column from the k-th.
gr>a=ara2aaa2b3dba2asa2ar, subtracts the s-th column from the r-th, and, at the same
time, adds the (p + r}th column to the (p + s)th.
g'r> s=ara2aga2&Q?&3a2aga2a,., adds the s-th column to the r-th, and, at the same time,
subtracts the (p + r)th from the (p + s)th column.
fr,s — ts9r,/at adds the (p + r)th and (p + s)th columns respectively to the s-th and
r-th columns.
f'r,a — tig'r,/»i subtracts the (^ + ?-)th and (p + s)th columns respectively from the s-th
and r-th columns.
To this list we add the matrix ak, whose effect has been described, and the matrix b'2,
which changes the sign both of the first and of the (p + l)th columns; then it is to be
shewn that a product, P, of positive integral powers of these matrices, can be chosen such
that, if A be any Abelian matrix of integers, given by
where aj8=/3a, a'£' = /3'a', a|3'-£a' = l,
the product AP is the matrix unity — of which every element is zero except those in the
diagonal, each of which is 1. Hence it will follow that fj. = P~1 ; namely that every such
Abelian matrix can be written as a product of positive integral powers of the matrices
Ak, B, C, D. Up to a certain point of the proof we shall suppose the matrix A to be
that for a transformation of any order, r.
In the matrices at, ar, aa, each of k, r, s is to be <jo + l; and in general each of
k, r, s is >1 ; but for the sake of uniformity it is convenient, as already stated, to
introduce a matrix ax = l ; then each of k, r, s may have any positive value less than p + l.
412]
ABELIAN MATRIX.
671
412. Of the matrix A we consider first the first row, and of this row we begin with
the jo-th and 2p-th elements, a,jp, 0lip ; if the numerically greater of these elements be
not a positive integer, use the matrix lp to make it positive*— form, that is, the product
Alp. Then, let y be the greater, and 8 the less of these two elements ; if 8 is positive,
use the matrix m'p or the matrix n'p, as many times as possible, to subtract from y the
greatest possible multiplef of 8 (i.e. if v be the matrix upon which we are operating, =A
or =&lp, form one of the products v(m'p)r, v (n'p)8) ; if 8 is negative, use mp or np to add
to y the greatest possible multiple of 8 ; so that, in either case, the remainder, y',
from y, is numerically less than 8 and positive. Now, by the matrix lpt take the element
8 to be positivej ; then again, by application of mp or np or m'p or n'p replace 8 by a
positive quantity numerically less than y'. Let this process alternately acting on the
remainder from y and 8, be continued until either y or 8 is replaced by zero. Then use
the matrix tp or l!v to put this zero element at the 2p-th place of the first row of the
matrix, A', which, after all these changes, replaces A.
Let a similar process of alternate reduction and transposition be applied to A', until
the (1, 2/>-l)th element of the resulting matrix is zero. And so on. Eventually we
arrive, in continuing the operation, at a matrix instead of A, in which there is a zero in
each of the places formerly occupied by /31(1, , 0lt p.
Now apply the processes given by b2, lp, gltp, gp>l, and eventually ap, if necessary, to
reduce the (1, p)th element to zero. Then the processes b2, lp,1, glt p_1, gp_ltl, ap_1, as
far as necessary, to reduce the (1, p-l)th element to zero; and so on, till the places,
which in the original matrix were occupied by a1>2, ..., alip, are all filled by zeros.
Consider now the second row of the modified matrix. Beginning with the (2, p)th and
(2, 2/?)th elements, use the specified processes to replace the latter by a zero. Next
replace, similarly, the (2, 2p-l)th element by a zero; and so on, finally replacing the
(2, jo + 2)th element by a zero. The necessary processes will not affect the fact that all
the elements in the first row, except the (1, l)th element, are zero. Next reduce the
elements occupying the (2, p)th, ..., (2, 3)th places to zero.
Proceeding thus we eventually have (i) the (r, s+p)th element zero, for every r<p and
every s<p, in which s>r, (ii) the (r, s)th element zero, for every r<p and every s<p, in
which s>r. In other words the matrix has a form which may be represented, taking p = 4,
by the matrix p,
p = ( an 0 0 0 0 0 0 0 );
«21
«22 °
0 021
0
0
0
«31
Q32 a33
0 031
032
0
0
«41
a42 a43
a44 041
042
043
0
«'ll
"12 «'i3
«14 #11
#12
#13
#14
«'«
a'42 a'43
a/44 0'41
0*42
0'43
0«
since now the original matrix is an Abelian matrix, and each of the matrices ak b c d is
an Abelian matrix, it follows (Chap. XVIII., § 324) that a0=0d ; if the original matrix be
* The changes of sign of the other elements of the same column which enter therewith do not
concern us.
t The simultaneous subtractions, effected by the matrix m'p, of the other elements of the
column, do not concern us. Similar remarks apply to following cases.
t It is not absolutely necessary to use the matrix /„ in this or in the former case ; but it con-
duces to clearness.
672 REDUCTION OF GENERAL [412
for greater generality supposed primarily to be associated with a transformation of order r,
the value r=l being introduced later, the determinant of the matrix is ±rf (§ 324, Ex. i.)
and is not zero ; hence comparing in turn the 1st, 2nd, ..., rows of the matrices a£ and £d
we deduce that in the matrix p the elements /821, /331, /332, ... of the matrix /3 which are on
the left side of the diagonal are also zero ; thus, in p, every element of the matrix £ is zero.
Apply now to the matrix p the relation
a/3'-/3d' = r,
which in this case reduces to aft = r. Then it is immediately found that the elements of
the matrix ft which are on the left side of the diagonal are also zero — and also that
an #11 = = appftpp=r.
The resulting form of the matrix p may then be shortly represented by
If now to the matrix a- we apply the processes given by the matrices glt 2 or g1^ 2 and £2,
we may suppose a21 numerically less than a^, and a22 positive ; if then we apply the
processes given by the matrices glt 3 or g'l> s and 13, and the processes given by the matrices
#2, 3 or #'2, s and ^3? we may suppose a31, a32 numerically less than a33, and may suppose a33
to be positive. Proceeding thus we may eventually suppose all the elements of any row of
the matrix a which are to the left of its diagonal to be less than the diagonal elements of
that row— and may suppose that all the elements of the diagonal of the matrix a are
positive ; this involves that the diagonal elements of ft are positive, and in particular
when r is a prime number involves that these elements are each 1 or r.
Further we may reduce the elements of the matrix a which are in the diagonal of
a', and those which are to the left of this diagonal, by means of the diagonal elements of
the matrix ft. We begin with the elements of the last row of a ; by means of the
processes given by the matrices np or rip we may suppose a'pp to be numerically less than
#PP 5 by means of the processes given by the matrices fp,p-l or f'p,p-i we may suppose
°'P,P-I to be numerically less than ftp,p; in general by means of the processes given by
fp,s or f'p,s we may suppose a'p>g to be numerically less than ftp,p. Similarly by the
processes given by »p_1 or n'p_1 we may suppose a'p_1>p_1 numerically less than ftp-llp-i,
and by the processes fp _ lt s or/'p_1)g, where s<p-l, we may suppose a'p_ljg numerically
less than ftp^lip_1. The general result is that in every row of the matrix a' we may
suppose the diagonal element, and the elements to the left of the diagonal, to be all
numerically less than the diagonal element of the same row of the matrix ft.
413. If then we take the case when r = l we have the result that it is possible to form
a product Q of the p + 2 matrices ak, b, c, d, such that the product AQ has a form which
may be represented, taking p = 3, by
A0 = ( 100000),
01 0 00 0
00 1 00 0
0 a'12 a'13 1 ft12 ftl3
0 0 a'23 0 1 #23
00 0 00 1
wherein all the elements of each of the matrices a and ft to the left of the diagonals are
zero, and all the elements of the matrix a both in the diagonal, and to the left of the
or ~
I TTNIVERSJ1
415]
ABELIAN MATRIX.
673
diagonal, are zero. Applying then the condition a/3' = l, we find that the elements of the
matrix /3' to the right of its diagonal are also zero, so that /3' = a= 1. Then finally, applying
the condition a ft' = ft a, equivalent to a' = a', we have a' = 0. Thus the reduced matrix is
the matrix unity of 2p rows and columns, and A, =O-1, is expressed as a product of
positive integral powers of the p+Z matrices Ak, B, C, D, as desired. Since the determinant
of each of the matrices Ak, B, C, D is +1, the determinant of the linear matrix A is also
+ 1.
414. In the particular case p = \ the only matrices of the p + 2 matrices Ak, B, C, D
which are not nugatory are the two matrices B and C ; we denote these here by U and V
and put further
u=U~l — ( \, v=V~l = ( ), vl = uvu3vu3, w = uvu3, w
then we immediately verify the facts denoted by the following table
M
&
UA
V
*t
10
»i
(-*£)
C-fe-9)
(«?, -£)
(l,i?+£)
Cfef-D
fc-%.t)
(5 + 7, >?)
of which, for example, the first entry means that if A = ( a, ^, ) be any matrix of 2 rows
\« P/
and columns, and we form the product Aw, then the columns £, 9 of the matrix A are
interchanged, and at the same time the sign of the new first column is changed ; we have
in fact
« /3W 01X/-/3 «
hence it is immediately shewn, as in the more general case, that every matrix A = ( a, ,} ,
\a PY
for which the integers a, /3, a', /3' satisfy the relation a£'-a'p" = l, can be expressed as a
product of positive integral powers of the two matrices
415. Combining the final result for the decomposition of a linear Abelian matrix with
the results obtained for any Abelian matrix of order r we arrive at the following statement,
whereof the parts other than the one which has been formally proved may be deduced from
~, ) be any Abelian matrix of order r ;
/
that one, or established independently : let A =
then it is possible to find a linear matrix Q expressible as a product of positive integral
powers of the (jo + 2) matrices Ak, B, C, D, which will enable us to write A = A;Q, where At
is an Abelian matrix of order r having any one, arbitrarily chosen, of the four forms repre-
sentable by
A
A'=
fc
and it is also possible to choose the linear matrix Q to put A into the form A = QAt, where
A< is also any one, arbitrarily chosen, of these same four forms. It follows that the deter
minant of the matrix A is +r". In virtue of the equations a«/3'« = r(t = l, ...,p~), which
hold for any one of the matrices A1} A2, A3, A4, and the inequalities which may also be
supposed to hold among the other elements, as exemplified, § 412, for the case of A^ it is easy
to find the number of different existing reduced matrices of any one of these forms. For
instance when p = 2, the number when r is a prime number is l+r + r^ + r5 ; for p = 3, and r
B. 43
674 LEMMAS IN REGARD TO GENERAL [415
a prime number, it is 1 +r + r* + 2r3 + ri+r5 + r6 ; for details the reader may consult Hermite,
Compt. Rendus, t. XL. (1855), p. 253, Wiltheiss, Crelle, xcvi. (1884), pp. 21, 22, and the
book of Krause, Die Transformation der Hyperelliptischen Functionen (Leipzig, 1886),
which deal with the case p = 2 ; for the case£> = 3, see Weber, Annali di Mat. Ser. 2", t. ix.
(1878), p. 139, where also the reduction to the form A = Q (r \ Q', in which Q, Q' are
linear matrices, is considered. Of. also Gauss, Disq. Arith., § 213 ; Eisenstein, Crelle, xxviu.
(1844), p. 327; Hermite, Crelle, XL., p. 264, XLI. (1851), p. 192; Smith, Phil. Trans. CLI.
(1861), Arts. 13, 14.
416. Considering (cf. § 372) any reduction, of the form
A
where ( p, °j is a linear matrix, we prove that however this reduction be effected, (i) the
determinant of the matrix B' is the same, save for sign, (ii) if p be a row of p positive
integers each less than r (including zero), the rows determined by the condition,
- -B'/i — integral, are the same. For any other reduction of this kind, say A=Q'A'0, must
be such that
,_p <r\ / q'-q\ _p q\ A B
~
where ( % ^ J is a linear matrix ; the condition that the matrix a' of the matrix A'0 should
vanish, namely p'A = 0, requires (since |4| 1^1=7* and therefore \A\, the determinant of
A, is not zero) that p' = 0 ; thus the reduction A = Q'A'0 can be written
(a p\ = /pq', -pq + <rp\ fpA , pB + qB'\
\ft.-ftj \p'q', -p'q + v'p) ' \0 , q'B' )'
Now pq1 = 1 ; therefore \q'\=±l ; thus \q'B'\= ±\B'\, which proves the first result. Also,
if fj. be a row of integers such that - B'p. is a row of integers, =m say, then - q'B'p, =q'm,
is also a row of integers ; while if -q'B'p be a row of integers, =n say, then -p
which is equal to -B'p, is equal to pn, and is also a row of integers ; since q'B' is the
matrix which, for the reduction A = Q'A'0, occupies the same place as that occupied, for the
reduction A = QA0, by the matrix B', the second result is also proved.
417. Considering any rectangular matrix whose constituents are integers, if all the
determinants of (£ + 1) rows and columns formed from this matrix are zero, but not all
determinants of I rows and columns, the matrix is said to be of rank I. The following
theorem is often of use, and is referred to § 397, Chap. XXII. ; In order that a system of
simultaneous not-homogeneous linear equations, with integer coefficients, should be capable
of being satisfied by integer values of the variables, it is necessary and sufficient that the
rank I of, and the greatest common divisor of all determinants of order I which can be
formed from, the matrix of the coefficients of the variables in these equations, should be
unaltered when to this matrix is added the column formed by the constant terms in these
equations. For the proof the reader may be referred to H. J. S. Smith, Phil. Trans. CLI.
(1861), Art. 11, and to Frobenius, Crelle, LXXXVI. (1879), pp. 171—2.
418. Consider a matrix of n+l columns and n+1 or more rows, whose constituents
are integers, of which the general row is denoted by
^f "i ...... ^'t'j 'tj ^i 5
419] MATRICES OF INTEGERS. 675
let A be the greatest common divisor of the determinants formed from this matrix with
n + 1 rows and columns; let A' be the greatest common divisor of the determinants
formed from this matrix with n rows and columns ; then, since every determinant of the
(n + l)th order may be written as a linear aggregate of determinants of the n-th order,
the quotient A/ A' is integral, = M, say. Then the n + l or more simultaneous linear
congruences
Ui = aix+biy+ ...... +&ie+lit + eju=Q (mod. M)
have just A incongruent sets of solutions, and have a solution whose constituents have unity as
their highest common divisor. Frobenius, Crelle, LXXXVI. (1879), p. 193.
Also, if in the m linear forms (m< = or >n + l)
Ui=aix+biy + ...... +kiz + lit + eiut (i=l, ...,m),
the greatest common divisor of the m(n + l) coefficients be unity, it is possible to determine
integer values of x,y, ...,t, u, such that the m forms have unity as their greatest common
divisor; in particular, when n=l, if the 2m numbers at, bt have unity as their greatest
common divisor, and the fyn(m-l) determinants aibj-ajbi be not all zero, it is possible to
find an integer x so that the m forms atx+bi have unity as their greatest common divisor.
Frobenius, loo. cit., p. 156.
419. The theorem of § 418 includes the theorem of § 357, p. 589 ; it also includes the
simple result stated § 383, p. 637, note. It also justifies the assumption made in § 397,
that the periods Q, Q' may be taken so that the simultaneous equations aotf -a'x=\,
bxf-b'x=Q can be solved in integers in such a way that the 2p elements rx-b, rod -b'
have unity as their greatest common divisor; assuming that r is not zero so that the
p (2p- 1) determinants a^j-afc, ai6/-a/6i, a^'-a-b- are not all zero, and that Q' has
been taken so that the 2p integers av, ..., ap, a/, ..., ap' have no common divisor other
than unity, the necessary and sufficient condition for the solution of the equations
ax' - a'x=\, bx1 - b'x=Q is (§ 417) that the greatest common divisor, say M, of the p (2p - 1)
binary determinants spoken of should divide each of the 2p integers b1} ...,bp'\ if this
condition is not already satisfied we may proceed as follows : find two coprime integers
(§ 418) which satisfy the 2p congruences
i =0, \bi+nai = 0 (mod. M), (*=1,
and thence two integers p, a- such that Ao--/*p = l ; put Q,± =\Q.' + p.Q, Q1 =
i = bi\+aifji, Ai — bip + aia; Bi=bl\ + ain, .4 / = 6^ + 0/0-; then
and the greatest common divisor of the p(2p-l} binary determinants AiBj-AjBi,
AiBj -Aj'Bi, A^Bj'- A-Bi, which is equal to M, divides the 2jo integers £lt ..., Bp';
thus M is the greatest common divisor of these 2p integers; next put Q2=MQl, Q^Q/,
bi = Bi/Af, \>i=BijM, &i = Ai, aj'=Ji'; then the greatest common divisor of the p(2p- 1)
binary determinants a^- — a,-bi, etc., is unity, and this is also the greatest common divisor
of the 2p integers bt, ..., bp'. Now let (x, x1} be any solution of the equations &x'-a'x=l,
b^-b'^=0, so that (rx — b, rx" -\j] is a solution of the equations a£' — a'£ = 0, b£'-b'£=0;
let (£» I') be an independent solution of these latter equations (Smith, Phil. Trans., CLI.
(1861), Art. 4) so that the p(2p-l) binary determinants #i&-#>&, etc., are not all zero,
so chosen that the 2p elements £,-, &' have unity as their highest common divisor; then if
h be any integer, the 2p elements #<+!!&, a?/ + h^' form a solution of the equations
ajcf-&'x=\, bo/-b'ar=0; let h be chosen so that the 2p elements rxt - b< + hr£,- ,
7vr/-b/ + hr£i' have no common factor greater than unity (§ 418). Putting A"=.r
676 COMPLETION OF DEFECTIVE ABELIAN MATRICES. [420
2T'=a/+h|', the first column of the matrix in § 397 will consist of the elements of (a, a'),
the (p + l)th column will consist of the elements of (b, b'), the second column will
consist of the elements of rJT-b, rJT'-b'; and since these latter have unity as their
greatest common factor, it is possible to construct the (^ + 2)th and all other columns
of this matrix (§ 420).
420. A theorem is assumed in § 396, which has an interest of its own— If of an
Abelian matrix of order r there be given the constituents of the first r columns, and also the
constituents of the (p + l)th, ..., (p + r}th columns (r<p\ it is always possible to determine
the remaining 2(p-r) columns. For a general enunciation the reader may refer to
Frobenius, Crette, LXXXIX. (1880), p. 40. We explain the method here by a particular case ;
suppose that of an Abelian matrix of order r, for p = 3, there be given the first an
columns ; denote the matrix by
( a x t
y? t'
b y u );
b' y" u' I
the elements of the given columns will satisfy the relation ab'-a'b=r ; it is required to
determine in order the second, the fifth, the third and the sixth columns ; the relations
arising from the equations
aa' — a'a = 0, ftft1 — ft' ft = 0, ap — a' ft = T
so far as they affect these columns respectively, are as follows :
ax'-a'x=Q\ . ay>-a'y = Q\ at'-a't = Q\ au'-a'u = (
bx'—b'x — O] by' — b'y=0\- (ii), bt' — b't = 0\ ..... bu' — b'u = (
I r (111)1 i /• \
xy'-yfy = r] xt-x't = Q\ xu'-afu = QY (iv);
yt — i/t = 0j yu' — y'u=Q\
. .
. . tu - tu = r>
now let (x, x1) be a solution of equations (i) in which the 2p constituents have no common
factor other than unity ; determine 2 rows of p elements £, £' such that xg — ofg = l, and
denote ag - a'£ by A and bg - 6'| by B ; then it is immediately verified that the values
y = r£-(Ab-Ba], T/ = rg - (AV - Ba"),
satisfy equations (ii) ; next let (t, t) be a solution of equations (iii) in which the 2p
constituents have no common factor other than unity ; determine 2 rows of p elements,
u, v, such that tv — t'v = l, and denote av'-a'v, bv'-b'v, xv'-x'v, yv'-y'v respectively by
A, B, X, T; then it is immediately verified that the values
u=rv-(Ab-Ba}- (Xy - Yx\ u1 = rv' -(Ab'~ Ba') -(Xy'- IV)
satisfy the equations (iv).
INDEX OF AUTHORS QUOTED. THE NUMBERS REFER
TO THE PAGES.
Abel 90, 173, 205, 206, 207 ff., 221, 225 ff.,
231, 243, 246, 377, 397, 657
Appell 200, 392
Bacharach 141
Bachmann 666
Baur 57, 112
Benoist 153, 156, 222, 647
Bertini 137
Biermann 663
Bobek 647
Bolza 177, 294, 296, 315, 329, 342, 436, 663
Borchardt 340, 342, 468
Bouquet 90
Braunmiihl 486, 521
Brill 12, 29, 134, 137, 145, 149, 436, 647
Brioschi 296, 311, 342, 448, 516, 526, 663, 666
Briot 90
Broch 221
Burkhardt 43, 429, 436, 555, 623, 663
Burwide 345, 373, 663
Cantor 239
Casorati 579
Caspary 474, 486, 525
Castelnuovo 647, 665
Cauchy 600
Cayley 12, 137, 141, 145, 165, 168, 193, 222,
230, 283, 296, 340, 342, 374, 387, 459,
540, 646, 647, 650, 664, 666, 668
Chasles 137, 647
Christo/el 666
Clebsch 131, 142, 153, 156, 165, 168, 183,
222, 241, 244, 288, 295, 392, 423, 448,
544, 545, 556, 578, 623, 647
Darboux 666
Dedekind 57
de Jonquieres 137
Dersch 647
Dini 239
Dirichlet 246, 600
Eisemtein 246, 674
Epstein 342
Euler 159
Fagnano 638
Forsyth 2, 3, 7, 9, 10, 13, 14, 15, 16, 21, 24,
25, 29, 39, 90, 114, 122, 123, 144, 150,
198, 212, 233, 296, 327, 373, 395, 421,
439, 441, 442, 445, 459, 531, 577, 578
Frahm 383
Fricke 639, 648
Frobenius 342, 387, 447, 474, 486, 491, 500,
516, 517, 521, 522, 525, 586, 588, 589,
598, 628, 629, 630, 632, 633, 666, 668,
674, 676
Frost 389
Fuchs 206, 566
Gauss 559, 600, 674
Geiser 383
Gopel 246, 338, 339
Gordan 131, 142, 168, 183, 241, 244, 255,
288, 295, 392, 423, 448, 544, 545, 556,
578, 623
Goursat 663
Grassman 137
Greenhill 639
Giinther 174, 189, 200
Halphen 124, 165, 364, 370, 421, 473, 474,
639, 665
Hamburger 2
Hancock 296, 326
Harkness 2, 10, 14, 15, 16, 21, 24, 25, 79,
101, 124, 239, 342
678
INDEX OF AUTHORS QUOTED.
Harnack 222
Hensel 57, 64, 78, 118
Hermite 238, 246, 448, 452, 538, 577, 600,
615, 632, 639, 663, 666. 674
Hettner 177, 654
Hilbert 665
Humbert 222, 255, 340, 486
Hunoitz 41, 392, 639, 648, 651, 653, 654
Jacobi 165, 206, 221, 230, 235, 237, 246, 360,
577, 600, 639, 657
Jordan 248, 392, 549, 623, 639, 666, 669
Joubert 639
Jiirgensen 221
Kiepert 638, 639
Klein 9, 25, 156, 159, 169, 177, 342, 343,
360, 373, 378, 383, 392, 429, 430, 431,
433, 436, 438, 439, 544, 639, 648, 653,
654
Kohn 387
Konigsberger 337, 342, 448, 459, 477, 529,
570, 600, 607, 615, 628, 658, 661, 663
Kowalevski 658, 659, 662, 663
Krause 296, 342, 468, 486, 600, 623, 674
Krazer 477, 486, 555, 600, 627
Kronecker 56, 79, 124, 445, 577, 600, 629,
631, 639, 666, 669
Kummer 340
Lagrange 230
Laguerre 632, 666
Lindemann 153, 156, 222, 647
Liiroth 239
Malet 663
Mathews 165
Minding 221
Hittag-Leffler 202
Morley 2, 10, 14, 15, 16, 21, 24, 25, 79, 101,
124, 239, 342
Netto 20, 90
Neumann 14, 17, 169, 296, 531
Noether 12, 29, 32, 124, 131, 134, 137, 142,
145, 149, 156, 165, 168, 180, 272, 292,
295, 390, 392, 430, 486, 522, 544, 566,
654, 665
Picard 14, 165, 594, 659
Pick 360, 430, 639
Plucker 124, 165
Poincare 239, 372, 373, 439, 486, 594, 654,
659, 663
Pringsheim 445
Prym 2, 296, 342, 392, 477, 486, 511, 566,
600, 627
Richelot 221, 230, 529, 600, 663
Riemann 1, 2, 6, 9, 13, 45, 47, 77, 113, 115,
246, 248, 255, 296, 343, 397, 409, 628
Ritter 360, 373, 392, 429, 439, 442
Roch 29
Rosanes 666
Rosenhain 221, 222, 246, 311, 459, 600, 607,
628, 662
Salmon 5, 6, 7, 11, 39, 117, 124, 136, 144,
159, 165, 267, 383, 389
Schepp 239
Sclilafli 666
Schottky 32, 101, 283, 296, 340, 343, 345-,
360, 371, 372, 373, 387, 448, 461, 469,
486, 544, 628
Schubert 665
Schwarz 14, 654
Scott 473
Smith 12, 600, 639, 666, 674, 675
Sohnke 639
Stahl 288, 301, 392, 430, 486, 502
Stickelberger 666
Stolz 2
Sylvester 136
Taber 668
Thomae 288, 296, 318, 533, 556, 600
Thompson 436
Toeplitz 383
Valentin 101
Valentiner 124, 165, 665
Voss 137
Weber 8, 56, 270, 272, 373, 387, 392, 430,
460, 486, 533, 538, 559, 600, 615, 620,
629, 639, 674
Weierstrass 32, 93, 99, 101, 177, 195, 197,
205, 231, 239, 242, 246, 301, 311, 317,
326, 339, 443, 474, 486, 525, 570, 571,
572, 573, 577, 579, 594, 628, 653, 654,
658, 666
White 165
Wiltheiss 342, 600, 629, 660, 662, 674
Wirtinger 340, 486, 578, 628
TABLE OF SOME FUNCTIONAL SYMBOLS.
Riemann's normal elementary integrals
of first kind, generally, v*'a, ... , v*' a, p. 15. For periods, p. 16,
of second kind, T*' a ; periods of, fy, ... , ftp, or Qa (z), ... , Op (z), pp. 15, 21,
of third kind, II*'", p. 15.
Integral, rational, functions, git or gt (x, y), or </; (y, x), pp. 55, 61.
0-polynomials, special functions, numerators of differential coefficients of integrals of the first
kind, 01} ..., #„_!, p. 61. Also 01( ..., 0P, p. 146.
Elementary integral of third kind, P£«, p. 68. (Canonical integral), Q*Ba, p. 185. (Canonical
integral), jR*'c", p. ]94.
Integrals of second kind, associated with given system of integrals of first kind, L*'a, p. 193;
periods of, 196. Also #*• ", p. 182, and F* °, p. 291, are used for integrals of second kind.
*(*, a? 2, CL ..., cp), pp. 77, 171, 177. This is called Weierstrass's fundamental rational
function.
$(x,a; z, c), pp. 174, 175, 178, 200.
E(x, z), pp. 171, 178 (Prime function).
E (x, z), pp. 176, 178, 205 (Prime function).
Matrices, see Appendix n., p. 666.
; <?,<?') ore(M,r QQ\ or Q ( u \%\ or 0(«; Q, Q>)
\ " / \ i V /
(«; Q, Q') or 3
o
), p. 287.
j\rt
Pi, i (u) = - ^—5^. Io8 M«). P- 292. See also p. 516.
Wi(x) (Differential coefficient of integral of first kind), p. 169. Also u- (r) p 192
„,-,_,, p. 192. ?,.,-, p. 288.
W(x, z; clt ..., <:„), p. 174.
™ (f. 7). P- 360 (Prime function). But for w (x, z), see pp. 430 428
\ (fc |t), p. 367.
| Q |, Q,R |, Qj).P.48f,
* (u, a ; J), p. 509.
0 (u), a Jacobian function, p. 579, ff.
^r(w; K, K' + fj.), ^r(ir; A', K' + /JL), p. 601.
SUBJECT INDEX TO THE PAGES OF THIS VOLUME.
Abelian functions, 236, 600, see Inversion ; in
tegrals, see Integrals; matrix, 669.
Abel's theorem, 207, ff.; statement of, 210,
214 ; proof of, 213 ; number of inde
pendent equations given by, 222 ff. ;
for radical functions, 377 ; for factorial
functions, 397 ; for curves in space,
231; Abel's proof of, 219, 220; con
verse of, 222.
Abel's differential equations, 225, ff.
Addition equation for hyperelliptic theta func
tions, deduced algebraically, 331, ff. ;
for theta functions in general, 457 —
461, 472, 476, 481, 513, 521.
Adjoint polynomial (or curve), definition of,
121 ; number of terms in, 128 ; ex
pression of rational function by, 127 ;
see Integrals, Sets, Lots.
Argument and parameter, interchange of, 16,
185, 187, 189, 191, 194, 206.
Associated : Forms associated with fundamental
integral functions, 62 ; integrals of
second kind associated with integrals
of the first kind, 193, 195, 198, 532 ;
associated system of factorial func
tions, 397
Automorphic functions, simple case of, 352, ff.;
connection with factorial functions,
439, ff.
Azygetic characteristics, 487, 497 ; transforma
tion of, 542, 547 ; see Characteristics.
Bacharach's modification of Cayley's theorem
for plane curves, 141.
Biquadratic, see Gopel.
Birational transformation of a Eiemann sur
face : does not affect the theory, 3, 7 ;
number of invariants in, 9, 144, 148,
150 ; of plane curves, 11 ; by 0-poly-
nomials, 142 — 152 ; for hyperelliptic
surface, 152, 85; when p — l, or 0,
153 ; of surface into itself, 653. See
Invariants, and Curves.
Bitangents of a plane curve, 381—390; 644, 646.
Branch places, see Places.
Canonical equation for a Kiemann surface, 83,
91, 103, 143, 145, 152; curve discussed
by Klein, 159 ; integral of the third
kind, 168, 185, 189, 194, 195.
Cayley's theorem for plane curves, 141.
Characteristics: of a theta function, number
of odd and even, 251; expression of
any half-integer characteristic by
means of a fundamental system, 301,
487, 500, 502; Weirstrass's number
notation for, 570, 337, 303 ; tables of
half -integer characteristics for p = 2,
p = 3, 303, 305; syzygetic, azygetic,
487; period characteristics and theta
characteristics, 543, 564; of radical
functions, 380, 564 ; Gopel groups
and systems of, 489, 490, 494, ff. ;
general theory of, 486, ff . ; transform
ation of, 536, 542, 547, 564, 568.
Coincidences of a correspondence, 645.
Column and row. See Matrices.
Column of periods, 571.
Complex multiplication of theta functions,
629, ff., 639, 660.
Composition of transformations of theta func
tions, 551.
Condition of dimensions, 49.
Confonnal representation, 343, 356, 372.
Congruence, meanings of sign of, 236, 256, 261,
264, 487.
Constants, invariant in rational transformation,
9, 88, 144, 148, 150 ; in linear trans
formation of theta functions, 555 —
559; in any transformation of theta
functions, 620, 622.
Contact curves, see Curves, and Radical.
SUBJECT INDEX TO THE PAGES OF THIS VOLUME.
681
Convergence of an automorphic series, 350;
of transformed theta function, 538.
Coresidual sets of places on a Riemann surface,
135, ff., 213; are equivalent sets, 136;
enter in statement of Abel's theorem,
210.
Correspondence of Riemann surfaces, 3, ff.,
81, 639, 642, 647, 648, 649, 654, 662.
Covariant, see Invariant.
Cubic surface associated with a plane quartic
curve, 382, 389.
Curves : as alternative interpretation of fun
damental algebraic equation, 11 ; in
flexions of a plane quartic in con
nection with the gap theorem, 36 ;
generalisation, 40 ; inflexions and
bitangents in connection with theory
of correspondence, 644, 646 ; bitaugents
of a plane quartic curve, 384 ; adjoint
curves, 121, 129 ; coresidual and
equivalent sets upon, 134 — 136 ; trans
formation of, see Birational, In
variants, and Constants ; correspon
dence of, see Correspondence ; special
sets upon, 146, ff.; contact curves,
381 ; general form of Pliicker's equa
tions for, 124 ; Weierstrass's canon
ical equation for, 93, 103 ; Cayley's
theorem for, 141 ; curves in space, 157,
160, ff., 166, 664; Abel's theorem for,
231.
Cusps, 11, 114.
Deficiency of a Riemann surface, 7, 55, 60.
Denning relation for theta functions, 443.
Definition equation of theta functions of general
order, 448.
Degenerate Abelian integrals, 657.
Dependence of the poles of a rational function,
27.
Differential equations of inversion problem,
'225, ff. ; of theta functions, see Ad
denda (p. xx).
Differentials of integrals of first kind, 25, 62,
67, 127, 169.
Dimension of an integral function, 48, ff., 55 ;
condition of dimensions, 49.
Discriminant of a fundamental set of integral
functions, 74, 101, 124.
Dissection of the Riemann surface, 26, 529,
253, 257, 569, 297, 550, 560.
Double points of a Riemann surface (or curve),
1, 2, 3, 11, 114; tangents of a plane
curve, 644, 646.
Elementary integrals, see Integrals.
Equivalence, meanings of sign of, 236, 256,
261, 264, 487.
Equivalent sets of places on a Riemann sur
face, 134, ff., 136, 213.
Essential factor of the discriminant, 60, 74, 124.
Existence theorems, algebraically deducible,
78 ; references, 14.
Expression of any rational function, 77, 176,
212 ; of fundamental integral func
tions, 105, ff. ; of half-integer charac
teristic by means of a fundamental
system, 301, 487, 500, 502.
Factorial functions, 392, ff.; definition of, 396;
which are everywhere finite, 399; ex
pressed by factorial integrals, 403 ;
expressed by fundamental factorial
function, 413; with fewest poles, 406;
used to express theta functions, 423,
426 ; connection with automorphic
functions, 439, ff.
Factorial integrals, 398 ; which are everywhere
finite, 399 ; fundamental, having only
poles, 408 ; simplified form of that
integral, 411 ; expression of factorial
function by means of that integral,
412.
Function, automorphic, 352, ff., 439, ff.; fac
torial, see Factorial ; integral, see
Rational, and Transcendental ; ^ func
tion, 292, 324, 333, 516 ; prime, 172,
177, 205, also 360, 363, 428 ; radical,
374, 390, 565 ; rational, see Rational ;
Theta, see Theta functions, and
Transformation ; I function, 287, 292,
320 ; see Fundamental rational.
Fundamental algebraical equation, 10, 113.
Fundamental rational function, Weierstrass's,
171, 175, 177, 178, ff., 182.
Fundamental set for the expression of rational
integral functions, 48, ff., 55, 56, 57,
105, ff.
Fundamental system of theta characteristics,
301, 487, 500, 502.
Gap theorem, 32, 34, 93, 174.
Geometrical investigations, 113; see Curves.
Go'pel biquadratic relation, 338 — 340 ; 465 —
468 ; see Addenda (p. xx).
Gopel group and system, see Characteristics.
Grade, of a polynomial, 120.
Group, Gopel, see Characteristics.
B.
44
682
SUBJECT INDEX TO THE PAGES OF THIS VOLUME.
Hensel's determination of fundamental integral
functions, 105, ff.
Homogeneous variables, 118, 441.
Homographic behaviour of differentials of in
tegrals of first kind, 26.
Hyperelliptic surfaces, 80, ff., 152, 153, 373;
see Theta functions and Transforma
tion.
Independence of the poles of a rational func
tion, 27; of the 22p theta functions
with half-integer characteristics, 446,
447 ; See Linearly.
Index of a place on a Riemann surface, 122,
123, 124; at the infinite place of
Weierstrass's canonical surface, 129.
Infinitesimal on a Riemann surface, 1, 2, 3.
Infinitesimal periods, 238, 573.
Infinities of rational function, 27, ff.; see
Residue.
Infinity, the places at infinity on a Riemann
surface, algebraic treatment of, 118.
Inflexions of a plane curve, 36, 40, 646.
Integrals, degenerate, 657; factorial, see Fac
torial; Riemann's, normal elementary,
15 ; all derivable from integral of third
kind, 22 ; algebraic expression of, 65,
ff., 127, 131, 163, 185, 189, 194;
hyperelliptic, 195 ; formulae connect
ing with logarithmic differential coeffi
cients of theta functions, 289, 290,320.
Integral functions, see Rational and Transcen
dental.
Interchange of argument and parameter, 16,
185, 187, 189, 191, 194, 206; of period
loops, see Transformation.
Invariants in birational transformation: the
number p, 7; the 3p-3 moduli, 9,
144, 148, 150; the ratios of ^-poly
nomials, 26, 153 ; the contact <f>-
polynomials, 281, 427; the 0-places,
38, 653; for transformation of the
dependent variable, 74, 124.
Inversion theorem, Jacobi's, 235, ff., 270;
solution of, 239, 242, 244, 275; by
radical functions, 390; in the hyper
elliptic case, 317, 324.
Jacobi's inversion theorem, see Inversion.
Jacobian functions, their periods, are generali
sation of theta functions, 579 — 588;
their expression by theta functions,
588 — 594 ; there exists a homogeneous
polynomial relation connecting any
p + 2 Jacobian functions of same
periods and parameter, 594.
Klein, prime form, 360, 427, 430, 433.
Laurent's theorem, for p variables, 444.
Left side of period loop, 529.
Linearly independent ^-products of order /u,
154; columns of periods, 575; theta
functions, 446, 447; Jacobian func
tions, 594.
Linear transformation, see Transformation.
Loops, period loops on a Riemann surface, 21,
529.
Lots, of sets of places on an algebraic curve,
or Riemann surface, 135.
Matrices, 248, 283, 580, 666, 669.
Mittag-Leffier's theorem for uniform function
on a Riemann surface, 202.
Moduli, of the algebraic equation, are 3p - 3 in
number, 9, 144, 148, 150; for the
hyperelliptic equation, 88.
Moduli of periodicity, see Periods.
Multiplication, complex, of theta functions,
629, ff. ; by an integer, for theta
functions, 527.
Multiply-periodic, 236 ; see Inversion.
Noether's (Kraus's) 0-curve in space, 156, 157.
Normal equation for a Riemann surface, 83,
91, 103, 143, 145, 152.
Normal integrals (Riemann's) see Integrals.
Number of independent products of /j. 0-poly-
nomials, 154; of odd and even theta
functions, 251 ; of theta functions of
general order, 452, 463; of Jacobian
functions, 594.
Order of small quantity on a Riemann surface, 2;
of a theta function, 448.
#> Function, 292, 324, 333, 516.
Parameter, interchange of argument and para
meter, see Interchange.
Parameters, in the algebraic equation, see
Constants.
Period loop, see Loops.
Period characteristics, see Characteristics.
Periodicity of a (1, 1) correspondence, 650.
Periods of Riemann's integrals, 16, 21 ; Rie
mann's and Weierstrass's relations for
the periods of integrals of the first
kind, and of associated integrals of
the second kind, 197, 285, 581, 587 ;
SUBJECT INDEX TO THE PAGES OF THIS VOLUME.
683
rule for half-periods on a hyperelliptic
surface, 297 ; for integrals of second
kind, 323 ; of factorial integrals, 404 ;
linear transformation of periods, 532 ;
general transformation, 536, 538 ;
general theory of systems of periods,
571, ff., 579, ft". ; of degenerate inte
grals, 657.
Picard's theorem (Weierstrass's), G58.
Places, of a lliemann surface, 1, 2, 3 ; branch
places, 7, 9, 46, 74, 122, 297, 569;
where a rational fiinction is infinite,
to order less than p + l, 38, 41, 90,
653;
the places mlt ..., m}> , 255 ; their geo
metrical interpretation, 265, 2G6 ; after
linear transformation, 562 ; deter
mination of, for a Riemann surface
with assigned period loops, 567 ; for a
hyperelliptic surface, 297, 563.
Plucker's equations, generalised form of, 123,
124 ; for curves in space, 166.
Poles, see Infinities.
Polynomial, grade of, 120 ; algebraic treat
ment of, 120; adjoint, 121, 128;
0-polynomials, 141 ; transformation
of fundamental equation by ^-polyno
mials, 142, 154 ; expression of rational
functions, and algebraic integrals by
means of adjoint polynomials, 156 ;
see Curves.
Positive direction of period loop, 529.
Primary and associated systems of factorial
functions, 397.
Prime function (or form), see Function.
Product expression of uniform transcendental
function with single essential singu
larity, 205.
Quartic. Double tangents of plane quartic
curve, 381—390, 647.
Quotients of theta functions, 310, 311, 390,
426, 516.
Radical function, see Function.
Rational function, of order 1, only exists when
p = 0, 8 ; is an uniform function on the
Riemann surface whose only infinities
are poles, 27 ; infinities of, Riemann -
Iloch theorem, Weierstrass's gap theo
rem, 27, ff. ; special, 25, 137 ; of order
p, 38, 137; integral function, 47, ff.,
55, 91, ff. ; of the second order, 80, ff. ;
fundamental integral rational func
tions, algebraic determination of, 105,
ff. ; algebraic expression of, by adjoint
polynomials, 125, ff., 156 ; Weier
strass's fundamental, 171, 175, 177,
178, ff., 182 ; expressed by Riemann's
integrals, 24, 212 ; expressed by Weier
strass's function, 176.
Reciprocal sets of zeros of adjoint polynomials,
134.
Residual sets of places, 135.
Residue, fundamental residue theorem, 232,
189, 20.
Reversible transformation, see Birational.
Riemann-Roch theorem, 44, 133 ; for factorial
functions, 405.
Riemann and Weierstrass's period relations,
197, 285, 581, 587.
Right side of period loop, 529.
Row and column, see Matrices
Schottky- Klein prime form and function, 360,
427, 430, 433.
Sequence, theorem of, 114, 161, 165.
Sequent sets of places, 135.
Sets of places on a Riemann surface or algebraic
curve, 135. See Special.
Sign of equivalence and congruence, 236, 256,
261, 264, 487.
Special correspondences on a Riemann surface,
648.
Special rational functions, 25, 62, 137.
Special sets of zeros of adjoint polynomials,
134, 147.
Special transformation of a theta function,
629, ff. , 639, 660.
Strength of assigned zeros, as determinators of
a polynomial, 133.
Supplementary transformations of a theta
function, 552.
System, Gopel, see Characteristics.
Syzygetic characteristics, 487, 542.
Tables of Characteristics, 303, 305.
Tangents, double, of a plane curve, by the
principle of correspondence, 644, 646.
Theta functions :
Riemann's theta functions, 246, ff. ; con
vergence of, 247; determination of,
from periodicity, 444 ; period proper
ties of, 249; number of odd and even,
251, 446; zeros of, 252, 255, 258, 567;
identical vanishing of, 258, 271, 276,
303; hyperelliptic, 296, ff. ; algebraic
expression of quotients of, 310, 311,
684
SUBJECT INDEX TO THE PAGES OF THIS VOLUME.
390, 426 ; addition theorem for hyper-
elliptic, 332, 337; algebraic expression
for hyperelliptic, 435; algebraic ex
pression of first logarithmic derivatives
of, 288, 290, 320; algebraic expression
of second logarithmic derivatives of,
293, 324, 329, 333 ; solution of inver
sion problem by means of, 275, 324,
390, 426, ff . ; Kiemann's functions not
the most general, 248, 628.
General theta function of first order, 283,
444; period relations, 285, 197, 581,
587; second logarithmic derivatives
of, 516; addition theorems for, 457,
472, 481, 513, 521 ; Gopel relation for,
in case p = 2, see Gopel; expression
of Jacobian functions by means of,
594.
Theta functions of second and higher order,
448 ; expression of, number of linearly
independent, 452, 463; of order 2, of
special kind, 509, 510; every p + 2
theta functions of same order, periods,
and characteristic, connected by a
homogeneous polynomial relation, 453.
Transformation of theta functions, see
Transformation ; characteristics of
theta functions, see Characteristics ;
complex multiplication of theta func
tions, 629, ff., 639, 660; theta func
tions expressed by factorial functions
and simpler theta functions, 426;
particular cases, 430, ff . ; hyperelliptic
case, 433.
Transcendental uniform function, 200 ; Mittag-
Leffler's theorem for, 202; expressed
in prime factors, 205; application of
Laurent's theorem when the function
is integral, 444.
Transformation
of the algebraic equation (or Kiemann
surface), 3, 143, 145, 151, 152, 654,
655 ; see Birational ;
of theta functions, 535; linear trans
formation, 539; constants in, 554 —
559; for hyperelliptic case, 568; of
second order, 603, 617; for any odd
order, general theorem, 614; con
stants in, 620, 622; when coefficients
not integers, 625 ; supplementary
transformations, 552 ; composition of,
551; special transformations, 629,
630, 660;
of periods, 528, 534, 539, 551, 553, 555,
559, 568;
of characteristics, see Characteristics.
Uniform, see Rational, and Transcendental.
Vanishing of theta function, 253, 258, 271 ff.,
276, 303.
Variables, homogeneous, 118, 429, 441 _
\ — /
Weierstrass's gap theorem, 32, 34, 93, 174;
special places which are the poles of
rational functions of order less than
p + l, 34, ff . ; canonical surface (or
equation), 90, ff., 93; fundamental
rational function, 171, 175, 177, 178,
182, 189; period relations, 197, ff.,
285, 581, 587; rule for characteristics
of hyperelliptic theta functions, 569 ;
theorem for degenerate integrals, 658.
Zeros, generalised zeros of a polynomial, 121 ;
zeros of Eiemann theta function,
252.
Zeta function, 287, 292, 320.
CAMBBIDGE : PRINTED BY J. AND C. F. CLAY, AT THE UNIVERSITY PRESS.
'
75
\1
UNIVERSITY OF CALIFORNIA LIBRARY
Live Music Archive
Librivox Free Audio
Metropolitan Museum
Cleveland Museum of Art
Internet Arcade
Console Living Room
Open Library
American Libraries
TV News
Understanding 9/11